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1.0.0 | problem:theoremdb-p9936 | problem | theoremdb-p9936 | Conjectured lower bound for solution-free subsets with respect to a linear equation | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $N$, let $[N]$ denote the set $\{1, 2, \ldots, N\}$. A subset $A \subseteq [N]$ is called solution-free if there do not exist distinct elements $x, y, z, w \in A$ satisfying the equation $x + 3y = 2z + 2w$. Let $f(N)$ denote the maximum cardinality of a solution-free subset of $[N]$. It is kno... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9936",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9936",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_16_conjectured_lower_bound :
∃ c > (0 : ℝ), (fun N ↦ (N : ℝ) * exp (-c * (log N) ^ (1 / 7 : ℝ))) ≪ fun N ↦ (f N : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_16_conjectured_lower_bound :\n ∃ c > (0 : ℝ), (fun N ↦ (N : ℝ) * exp (-c * (log N) ^ (1 / 7 : ℝ))) ≪ fun N ↦ (f N : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9936",
"format": "declaration-or-fragment",
"sha256": "a... | [] | TheoremDB P9936 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9936",
"url": "https://theoremdb.org/statements/P9936",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9936",
"not... | {
"source_record": "P9936",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9936,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.573Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c1b73647-b963-465c-be3b-0a3336e532c9","format_issues":null,"number":3889} |
1.0.0 | problem:theoremdb-p9938 | problem | theoremdb-p9938 | Asymptotic order of the iterated sum-of-divisors primality threshold | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n > 1$, let $c(n)$ denote the minimum number of iterations of the map $f(n) = \sigma(n) - 1$ required to reach a prime number, where $\sigma(n)$ denotes the sum of all positive divisors of $n$. That is, $c(n)$ is the least non-negative integer $i$ such that $f^{(i)}(n)$ is prime, where $f^{(i)}$ denot... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9938",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9938",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.variants.sigma_isTheta (c : ℕ → ℕ)
(h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
(fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma_isTheta (c : ℕ → ℕ)\n (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :\n (fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9938",
"form... | [] | TheoremDB P9938 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9938",
"url": "https://theoremdb.org/statements/P9938",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9938",
"not... | {
"source_record": "P9938",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9938,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.364Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"63ff0c35-decf-48b9-95bf-53aa47164f65","format_issues":null,"number":3404} |
1.0.0 | problem:theoremdb-p9940 | problem | theoremdb-p9940 | Base-3 Digits of Powers of Two | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For every integer $n > 8$, does the base-3 representation of $2^n$ contain the digit 2? In other words, for every integer $n > 8$, is $2^n$ not representable as a sum of distinct powers of 3? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9940",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9940",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem CollatzLike (n : ℕ) (hn : 8 < n) : 2 ∈ Nat.digits 3 (2^n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem CollatzLike (n : ℕ) (hn : 8 < n) : 2 ∈ Nat.digits 3 (2^n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9940",
"format": "declaration-or-fragment",
"sha256": "e2137aadbeb60c780648b72a478c03429164f019025a5a45852965a8efd3ad10",
... | [] | TheoremDB P9940 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9940",
"url": "https://theoremdb.org/statements/P9940",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9940",
"not... | {
"source_record": "P9940",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9940,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.609Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"61f5dfb6-04ed-4441-8d20-4da257b4e6ba","format_issues":null,"number":3453} |
1.0.0 | problem:theoremdb-p9942 | problem | theoremdb-p9942 | Lehmer's conjecture on the Ramanujan τ-function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\Delta$ be the formal power series defined by $\Delta = X \prod_{n=1}^{\infty} (1 - X^n)^{24}$ in $\mathbb{Z}[[X]]$, and let $\tau(n)$ denote the coefficient of $X^n$ in $\Delta$ for each non-negative integer $n$. Prove that for every positive integer $n$, $\tau(n) \neq 0$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9942",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9942",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lehmer_ramanujan_tau : ∀ n > 0, τ n ≠ 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lehmer_ramanujan_tau : ∀ n > 0, τ n ≠ 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9942",
"format": "declaration-or-fragment",
"sha256": "2f3fda3b8ddfbff5f94ea8b5156254b4c5ecf0562936d056c9be7dc2fca236b4",
"license": "Select... | [] | TheoremDB P9942 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9942",
"url": "https://theoremdb.org/statements/P9942",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9942",
"not... | {
"source_record": "P9942",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9942,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.152Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d26b1094-eff7-4695-bb62-1cc3f3df21e2","format_issues":null,"number":6034} |
1.0.0 | problem:theoremdb-p9944 | problem | theoremdb-p9944 | Characterization of pairs yielding additively complete floor-exponential sequences | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A pair of positive real numbers $(t, \alpha)$ is called a good pair if the sequence $\lfloor t\alpha^n\rfloor$ for $n = 0, 1, 2, \dots$ is additively complete, meaning that every sufficiently large positive integer can be expressed as a sum of distinct terms from this sequence. Determine the exact set of all good pairs... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9944",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9944",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_349 :
{(t, α) | 0 < t ∧ 0 < α ∧ IsGoodPair t α} = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_349 :\n {(t, α) | 0 < t ∧ 0 < α ∧ IsGoodPair t α} = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9944",
"format": "declaration-or-fragment",
"sha256": "2795f6a10746827673e3850580d054bb32e2eab07a6897a37fed57... | [] | TheoremDB P9944 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9944",
"url": "https://theoremdb.org/statements/P9944",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9944",
"not... | {
"source_record": "P9944",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:47+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9944,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.088Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dd33071e-1bcb-45e3-9b55-1bbcd1504a55","format_issues":null,"number":3558} |
1.0.0 | problem:theoremdb-p9946 | problem | theoremdb-p9946 | Convex Moser's Worm Problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A worm is defined as the range of a $1$-Lipschitz function from the interval $[0,1]$ to the plane $\mathbb{R}^2$, which represents a curve of length at most $1$. A worm cover is a measurable subset $X$ of $\mathbb{R}^2$ such that for every worm $w$, there exists an orientation-preserving rigid motion (a composition of ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9946",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9946",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem convex_mosers_worm_problem :
IsGLB {v | ∃ X ∈ WormCovers, Convex ℝ X ∧ volume X = v} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem convex_mosers_worm_problem :\n IsGLB {v | ∃ X ∈ WormCovers, Convex ℝ X ∧ volume X = v} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9946",
"format": "declaration-or-fragment",
"sha256": "e294de9d4c70837898dce6b4c... | [] | TheoremDB P9946 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9946",
"url": "https://theoremdb.org/statements/P9946",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9946",
"not... | {
"source_record": "P9946",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9946,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.712Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"78bcbdbd-1f14-4ec1-9766-7d74ddb4de6f","format_issues":null,"number":3919} |
1.0.0 | problem:theoremdb-p9948 | problem | theoremdb-p9948 | Same Parity Betrothed Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Two natural numbers $m$ and $n$ are called betrothed (or quasi-amicable) if the sum-of-divisors function $\sigma$ satisfies $\sigma(m) = \sigma(n) = m + n + 1$, where $\sigma(k)$ denotes the sum of all positive divisors of $k$. Equivalently, the sum of the proper divisors of $m$ equals $n + 1$, and the sum of the prope... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9948",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9948",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem same_parity_betrothed :
answer(sorry) ↔ ∃ m n : ℕ, m ≠ n ∧ IsBetrothed m n ∧ (Even m ↔ Even n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem same_parity_betrothed :\n answer(sorry) ↔ ∃ m n : ℕ, m ≠ n ∧ IsBetrothed m n ∧ (Even m ↔ Even n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9948",
"format": "declaration-or-fragment",
"sha256": "09a8f42467311d428ff2c981d4718... | [] | TheoremDB P9948 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9948",
"url": "https://theoremdb.org/statements/P9948",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9948",
"not... | {
"source_record": "P9948",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9948,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.713Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8ac3d8b7-d2d0-4691-bb3a-73cc65e6eb64","format_issues":null,"number":6587} |
1.0.0 | problem:theoremdb-p9950 | problem | theoremdb-p9950 | Erdős's problem on infinite chromatic number and short odd cycles | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathfrak{m}$ be an infinite cardinal. Suppose $G$ is a simple graph whose chromatic number equals $\mathfrak{m}$. For every natural number $r \geq 1$, must there exist a subgraph $H$ of $G$ such that the chromatic number of $H$ equals $\mathfrak{m}$ and $H$ contains no odd cycle of length at most $r$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9950",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9950",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_740 :
answer(sorry) ↔
∀ (V : Type*) (G : SimpleGraph V),
ℵ₀ ≤ G.chromaticCardinal →
∀ (r : ℕ),
∃ (H : G.Subgraph), H.coe.chromaticCardinal = G.chromaticCardinal ∧
NoShortOddCycle H.coe r | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_740 :\n answer(sorry) ↔\n ∀ (V : Type*) (G : SimpleGraph V),\n ℵ₀ ≤ G.chromaticCardinal →\n ∀ (r : ℕ),\n ∃ (H : G.Subgraph), H.coe.chromaticCardinal = G.chromaticCardinal ∧\n NoShortOddCycle H.coe r",
"target"... | [] | TheoremDB P9950 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9950",
"url": "https://theoremdb.org/statements/P9950",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9950",
"not... | {
"source_record": "P9950",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9950,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.478Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"75524195-e133-4f2c-90aa-4fa93d6a0feb","format_issues":null,"number":4749} |
1.0.0 | problem:theoremdb-p9952 | problem | theoremdb-p9952 | De Giorgi's conjecture in dimension five | {
"main": "math",
"sub": "partial differential equations",
"slug": "partial-differential-equations",
"name": "Partial differential equations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "35",
"basis": "community-map",
"evidence": "Inferred from community field: partial-differential-equations"
}
] | Let $n = 5$. Consider a function $u : \mathbb{R}^n \to \mathbb{R}$. We say that $u$ is a bounded classical solution to $\Delta u + u - u^3 = 0$ if $u$ is twice continuously differentiable, there exists a constant $C$ such that $|u(x)| \leq C$ for all $x \in \mathbb{R}^n$, and $\Delta u(x) + u(x) - u(x)^3 = 0$ for all $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9952",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9952",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem DeGiorgi_five : (DeGiorgi_conclusion 5) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_five : (DeGiorgi_conclusion 5)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9952",
"format": "declaration-or-fragment",
"sha256": "db29722ecbb9eda4b82351be6dfad1df54c4864366ccab856a3638b49f470d38",
"license": "Select... | [] | TheoremDB P9952 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9952",
"url": "https://theoremdb.org/statements/P9952",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9952",
"not... | {
"source_record": "P9952",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9952,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.905Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f716be71-f922-440e-8785-1c9d32bb6c0b","format_issues":null,"number":3961} |
1.0.0 | problem:theoremdb-p9954 | problem | theoremdb-p9954 | Distinct prime divisors for blocks of consecutive composite integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $n \geq 1$ and every integer $k \geq 0$, suppose that each of the consecutive integers $n+1, n+2, \ldots, n+k$ is composite. Does there always exist a sequence of distinct primes $p_1, p_2, \ldots, p_k$ such that $p_i$ divides $n+i$ for all $1 \leq i \leq k$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9954",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9954",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_375 : answer(sorry) ↔ Erdos375Prop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_375 : answer(sorry) ↔ Erdos375Prop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9954",
"format": "declaration-or-fragment",
"sha256": "e05dc3f0d52f06d16a61ae7033ac1ce428e4ffd2e67da03951f363d95a3aaece",
"license": "Selec... | [] | TheoremDB P9954 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9954",
"url": "https://theoremdb.org/statements/P9954",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9954",
"not... | {
"source_record": "P9954",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9954,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.240Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"713c2cdf-8e07-4de6-a9a1-75f844cb9a44","format_issues":null,"number":4033} |
1.0.0 | problem:theoremdb-p9956 | problem | theoremdb-p9956 | Catalan's Mersenne Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | The Catalan–Mersenne numbers are defined recursively by $c_0 = 2$ and $c_{n+1} = 2^{c_n} - 1$ for $n \geq 0$. The first five values $c_0, c_1, c_2, c_3, c_4$ are known to be prime. Determine whether every Catalan–Mersenne number $c_n$ with $n \geq 5$ is prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9956",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9956",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem catalans_mersenne_conjecture :
answer(sorry) ↔ ∀ n ≥ 5, Nat.Prime (catalanMersenne n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem catalans_mersenne_conjecture :\n answer(sorry) ↔ ∀ n ≥ 5, Nat.Prime (catalanMersenne n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9956",
"format": "declaration-or-fragment",
"sha256": "ad4db924d8d53a9860eb12ccd652d5c9216cbd... | [] | TheoremDB P9956 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9956",
"url": "https://theoremdb.org/statements/P9956",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9956",
"not... | {
"source_record": "P9956",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:54+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9956,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.013Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fe3eb8ba-d76d-4d10-9ad9-0cb6234c760c","format_issues":null,"number":3542} |
1.0.0 | problem:theoremdb-p9958 | problem | theoremdb-p9958 | Asymptotic growth of the smallest maximal Sidon set | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $N$, let $m(N)$ denote the minimum cardinality of a maximal Sidon set contained in $\{1, 2, \ldots, N\}$. A set $A \subseteq \{1, 2, \ldots, N\}$ is a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. A Sidon set $A \subseteq \{1, 2, \ldots, N\}$ is maximal if it is n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9958",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9958",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_156 :
answer(sorry) ↔
(fun N ↦ (minMaximalSidonSet N : ℝ)) =O[atTop] (fun N ↦ (N : ℝ) ^ (1 / 3 : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_156 :\n answer(sorry) ↔\n (fun N ↦ (minMaximalSidonSet N : ℝ)) =O[atTop] (fun N ↦ (N : ℝ) ^ (1 / 3 : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9958",
"format": "declaration-or-fragment",
"sha256": "61ec9aaa3e... | [] | TheoremDB P9958 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9958",
"url": "https://theoremdb.org/statements/P9958",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9958",
"not... | {
"source_record": "P9958",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9958,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.257Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bd158cac-7aa9-4a55-b87d-230e4765c4ff","format_issues":null,"number":3383} |
1.0.0 | problem:theoremdb-p9960 | problem | theoremdb-p9960 | The Lander–Parkin–Selfridge conjecture on equal sums of distinct powers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $k$, $n$, and $m$ be positive integers. Let $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_m$ be positive integers such that every $x_i$ is distinct from every $y_j$. Suppose that
$$
\sum_{i=1}^{n} x_i^k = \sum_{j=1}^{m} y_j^k.
$$
Must it follow that $k \leq n + m$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9960",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9960",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lander_parkin_selfridge :
∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),
0 < n → 0 < m →
(∀ i, 0 < x i) → (∀ j, 0 < y j) →
(∀ i j, x i ≠ y j) →
∑ i, x i ^ k = ∑ j, y j ^ k →
k ≤ n + m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lander_parkin_selfridge :\n ∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),\n 0 < n → 0 < m →\n (∀ i, 0 < x i) → (∀ j, 0 < y j) →\n (∀ i j, x i ≠ y j) →\n ∑ i, x i ^ k = ∑ j, y j ^ k →\n k ≤ n + m",
"target": null,
"source_url": "ht... | [] | TheoremDB P9960 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9960",
"url": "https://theoremdb.org/statements/P9960",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9960",
"not... | {
"source_record": "P9960",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9960,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.593Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4ce731c1-ccc0-4d72-9b2b-bd9de30bdb2d","format_issues":null,"number":6776} |
1.0.0 | problem:theoremdb-p9962 | problem | theoremdb-p9962 | Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to repr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9962",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9962",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_18a : answer(sorry) ↔
∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧
(practicalH m : ℝ) < (log (log m)) ^ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_18a : answer(sorry) ↔\n ∃ C : ℝ, 0 < C ∧ ∃ᶠ m in atTop, Nat.IsPractical m ∧\n (practicalH m : ℝ) < (log (log m)) ^ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P9962",
"format": "declaration-or-fragment",
"sha256":... | [] | TheoremDB P9962 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9962",
"url": "https://theoremdb.org/statements/P9962",
"note": null,
"sort_order": 0
},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9962",
"not... | {
"source_record": "P9962",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9962,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.347Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ddec9ce3-8a44-4f75-9a9b-1bcaf399c449","format_issues":null,"number":4721} |
1.0.0 | problem:theoremdb-p9964 | problem | theoremdb-p9964 | The P versus NP Problem | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | Let a decision problem be a function from lists of booleans to booleans. A complexity class is a set of decision problems. A function is computable in polynomial time if there exists a deterministic Turing machine that computes it in time polynomial in the input length. The complexity class P is the set of decision pro... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9964",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9964",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem P_ne_NP : P ≠ NP | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem P_ne_NP : P ≠ NP",
"target": null,
"source_url": "https://theoremdb.org/statements/P9964",
"format": "declaration-or-fragment",
"sha256": "c98375b7a7a081c474736668e7094bd57a5e6f35c1e8bbcd2615def78a838018",
"license": "Selected packets label their ... | [] | TheoremDB P9964 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9964",
"url": "https://theoremdb.org/statements/P9964",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9964",
"not... | {
"source_record": "P9964",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:58+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9964,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.689Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5674f0db-2c3d-4b7e-ba4e-79ee382620fe","format_issues":null,"number":6797} |
1.0.0 | problem:theoremdb-p9966 | problem | theoremdb-p9966 | Dubner's conjecture on sums of twin primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A natural number $p$ is called a twin prime if $p$ is prime and at least one of $p-2$ or $p+2$ is also prime. Prove or disprove: for every even natural number $n$ with $n > 4208$, there exist twin primes $p$ and $q$ such that $p + q = n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9966",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9966",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem dubner_conjecture (n : ℕ) (hn : 4208 < n) (h : Even n) :
∃ p q : ℕ,
IsTwinPrime p ∧
IsTwinPrime q ∧
p + q = n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem dubner_conjecture (n : ℕ) (hn : 4208 < n) (h : Even n) :\n ∃ p q : ℕ,\n IsTwinPrime p ∧\n IsTwinPrime q ∧\n p + q = n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9966",
"format": "declaration-or-fragment",
"sha... | [] | TheoremDB P9966 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. Not Lean-verified
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9966",
"url": "https://theoremdb.org/statements/P9966",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9966",
"not... | {
"source_record": "P9966",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:59+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9966,"public_status":{"evidence":{"hasFormalization":true,"hasInformalProof":false,"hasRecordedWork":true,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"Not Lean-v... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:54.700Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"85c37b03-3072-447d-a4b3-c57d05a59638","format_issues":null,"number":4359} |
1.0.0 | problem:theoremdb-p9968 | problem | theoremdb-p9968 | Unique representation by sums with monomial values | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $k \ge 2$ be a natural number and let $f(x) = x^k$. For any set $A \subseteq \mathbb{Z}$, let $f(\mathbb{Z}_{>0}) = \{f(n) : n \in \mathbb{Z}, n > 0\}$ denote the set of positive-integer evaluations of $f$. For each integer $z$, consider whether there exist unique elements $a \in A$ and $b \in f(\mathbb{Z}_{>0})$ s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9968",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9968",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_477.variants.monomial (k : ℕ) (hk : 2 ≤ k) :
letI f := X ^ k
∀ A : Set ℤ, ∃ z, ¬ ∃! a ∈ A ×ˢ (Set.range f.eval), z = a.1 + a.2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_477.variants.monomial (k : ℕ) (hk : 2 ≤ k) :\n letI f := X ^ k\n ∀ A : Set ℤ, ∃ z, ¬ ∃! a ∈ A ×ˢ (Set.range f.eval), z = a.1 + a.2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9968",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9968 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9968",
"url": "https://theoremdb.org/statements/P9968",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9968",
"not... | {
"source_record": "P9968",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9968,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.237Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5203ce1d-2e00-4610-86d6-99399f0cd879","format_issues":null,"number":6916} |
1.0.0 | problem:theoremdb-p9970 | problem | theoremdb-p9970 | Asymptotic strict upper bound for minimum arithmetic-progression covering sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each pair of natural numbers $N$ and $k$, let $m(N,k)$ denote the minimum cardinality of a subset $A \subseteq \mathbb{N}$ with the property that for every $d \in \{1, 2, \ldots, N\}$, the set $A$ contains an arithmetic progression of length $k$ with common difference $d$. For a fixed positive integer $k$, determin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9970",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9970",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_37_littleO (k : ℕ) :
(fun N ↦ (m N k : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_37_littleO (k : ℕ) :\n (fun N ↦ (m N k : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9970",
"format": "declaration-or-fragment",
"sha256": "c336b762d263f1f4c0a526eb8c147d74ae151d4c6... | [] | TheoremDB P9970 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9970",
"url": "https://theoremdb.org/statements/P9970",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9970",
"not... | {
"source_record": "P9970",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9970,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.396Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c0c5341b-3926-442a-b101-6c9cb4b4dcc9","format_issues":null,"number":3411} |
1.0.0 | problem:theoremdb-p9972 | problem | theoremdb-p9972 | Asymptotic growth of the maximum guaranteed order of a regular induced subgraph | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, let $F(n)$ denote the largest integer such that every finite simple graph on $n$ vertices contains a regular induced subgraph with at least $F(n)$ vertices. Here, a regular induced subgraph of a graph $G$ is an induced subgraph that is $k$-regular for some non-negative integer $k$, meanin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9972",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9972",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_82 : Tendsto (fun n => F n / Real.log n) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_82 : Tendsto (fun n => F n / Real.log n) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9972",
"format": "declaration-or-fragment",
"sha256": "176c4dc24ddce90e66fea4702cbd3699cb78975fc0ff283b57492ffd07a1e9b9",
... | [] | TheoremDB P9972 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9972",
"url": "https://theoremdb.org/statements/P9972",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9972",
"not... | {
"source_record": "P9972",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9972,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.221Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0c86d8e3-e9c1-43e6-82be-5435187da94a","format_issues":null,"number":3376} |
1.0.0 | problem:theoremdb-p9974 | problem | theoremdb-p9974 | Erdős's Egyptian Fraction Decomposition for Four-Nths | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $n > 2$, determine whether there exist distinct natural numbers $x$, $y$, and $z$ with $1 \leq x < y < z$ such that
$$
\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
$$
. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9974",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9974",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_242 (n : ℕ) (hn : 2 < n) :
∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧
(4 / n : ℚ) = 1 / x + 1 / y + 1 / z | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_242 (n : ℕ) (hn : 2 < n) :\n ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧\n (4 / n : ℚ) = 1 / x + 1 / y + 1 / z",
"target": null,
"source_url": "https://theoremdb.org/statements/P9974",
"format": "declaration-or-fragment",
"sha256": "e188ee0b817... | [] | TheoremDB P9974 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9974",
"url": "https://theoremdb.org/statements/P9974",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9974",
"not... | {
"source_record": "P9974",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9974,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.383Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d5f94bbe-d48d-4ef2-bc5d-387df47aa0a8","format_issues":null,"number":4729} |
1.0.0 | problem:theoremdb-p9976 | problem | theoremdb-p9976 | Existence of a Continuous Density Function for Scaled Prime Gaps | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $p(n)$ denote the $n$th prime number, and define the prime gap $\operatorname{gap}(n) = p(n+1) - p(n)$. For a real number $c \geq 0$, let $S_c = \{n \in \mathbb{N} : \operatorname{gap}(n) / \log n < c\}$. We say that a set $A \subseteq \mathbb{N}$ has density $d$ if the limit $\lim_{N... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9976",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9976",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_234 : answer(sorry) ↔ ∃ f : ℝ≥0 → ℝ, Continuous f ∧
∀ c : ℝ≥0, HasDensity {n : ℕ | primeGap n / log n < c} (f c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_234 : answer(sorry) ↔ ∃ f : ℝ≥0 → ℝ, Continuous f ∧\n ∀ c : ℝ≥0, HasDensity {n : ℕ | primeGap n / log n < c} (f c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9976",
"format": "declaration-or-fragment",
"sha256": "e7fbc... | [] | TheoremDB P9976 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9976",
"url": "https://theoremdb.org/statements/P9976",
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{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9976",
"not... | {
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9976,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.412Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30c16d1c-ceca-4c3c-a2b6-dd1b09209787","format_issues":null,"number":5177} |
1.0.0 | problem:theoremdb-p9978 | problem | theoremdb-p9978 | Finite Implication from Equation 677 to Equation 255 in Magmas | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A magma is a set $G$ equipped with a binary operation, denoted by $\circ$. For a magma $G$, define the following equations:
- Equation 677: for all $x, y \in G$, $x = y \circ (x \circ ((y \circ x) \circ y))$.
- Equation 255: for all $x \in G$, $x = ((x \circ x) \circ x) \circ x$.
Let $G$ be a finite magma. Prove that... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9978",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9978",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem Finite.Equation677_implies_Equation255 (G : Type) [Magma G] [Finite G]
(h : Equation677 G) : Equation255 G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Finite.Equation677_implies_Equation255 (G : Type) [Magma G] [Finite G]\n (h : Equation677 G) : Equation255 G",
"target": null,
"source_url": "https://theoremdb.org/statements/P9978",
"format": "declaration-or-fragment",
"sha256": "a4b41cfaf15828e23... | [] | TheoremDB P9978 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9978",
"url": "https://theoremdb.org/statements/P9978",
"note": null,
"sort_order": 0
},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9978",
"not... | {
"source_record": "P9978",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9978,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.005Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"74296950-3e20-4f27-b75e-1ed22f748479","format_issues":null,"number":5302} |
1.0.0 | problem:theoremdb-p9980 | problem | theoremdb-p9980 | The Happy Ending Problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each integer $n \geq 3$, let $f(n)$ denote the smallest integer $N$ such that every set of $N$ points in the plane $\mathbb{R}^2$, with no three points collinear, contains $n$ points that are the vertices of a convex $n$-gon. Determine whether $f(n) = 2^{n-2} + 1$ holds for all $n \geq 3$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9980",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9980",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9980",
"format": "declaration-or-fragment",
"sha256": "4e1680eec1bd60b768c307e1f2826e74c35b7c52678bd0e93efc678df5570745",
"... | [] | TheoremDB P9980 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9980",
"url": "https://theoremdb.org/statements/P9980",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9980",
"not... | {
"source_record": "P9980",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:11+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9980,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.529Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bcb69ba8-1f66-4762-af36-ce735dda2339","format_issues":null,"number":6762} |
1.0.0 | problem:theoremdb-p9982 | problem | theoremdb-p9982 | Greedy algorithm termination for residue-class Egyptian fractions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $x$ be a rational number with $0 < x < 1$. Let $a$ and $d$ be natural numbers with $d > 1$. Assume that $\gcd\left(\frac{\mathrm{den}(x)}{\gcd(\mathrm{den}(x), \gcd(a,d))}, \frac{d}{\gcd(a,d)}\right) = 1$, where $\mathrm{den}(x)$ denotes the denominator of $x$ in lowest terms. Consider the greedy algorithm for Egyp... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9982",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9982",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_282.variants.graham {x : ℚ} (hx : x ∈ Set.Ioo 0 1) {a d : ℕ} (hd : 1 < d)
(h : (x.den / x.den.gcd (a.gcd d)).gcd (d / a.gcd d) = 1) :
(greedyUnitFractionRem { n | n ≡ a [MOD d] } x =ᶠ[atTop] 0) ↔ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_282.variants.graham {x : ℚ} (hx : x ∈ Set.Ioo 0 1) {a d : ℕ} (hd : 1 < d)\n (h : (x.den / x.den.gcd (a.gcd d)).gcd (d / a.gcd d) = 1) :\n (greedyUnitFractionRem { n | n ≡ a [MOD d] } x =ᶠ[atTop] 0) ↔ answer(sorry)",
"target": null,
"source_url": "... | [] | TheoremDB P9982 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9982",
"url": "https://theoremdb.org/statements/P9982",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9982",
"not... | {
"source_record": "P9982",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9982,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.411Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"48a6735b-9607-482a-bebc-e19336b453de","format_issues":null,"number":5389} |
1.0.0 | problem:theoremdb-p9984 | problem | theoremdb-p9984 | Erdős's bipartization bound for triangle-free graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a natural number $n$, let $V$ be a finite set with $|V| = 5n$. Let $G$ be a simple graph on $V$ that contains no triangle (that is, $G$ is $K_3$-free). Determine whether there always exists a bipartite subgraph $H$ of $G$ such that at most $n^2$ edges of $G$ are not edges of $H$. Equivalently, can every triangle-fr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9984",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9984",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_23 : answer(sorry) ↔
∀ (n : ℕ) (V : Type) [Fintype V], Fintype.card V = 5 * n →
∀ (G : SimpleGraph V), G.CliqueFree 3 →
∃ (H : SimpleGraph V),
H ≤ G ∧ H.IsBipartite ∧ (G.edgeFinset \ H.edgeFinset).card ≤ n^2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_23 : answer(sorry) ↔\n ∀ (n : ℕ) (V : Type) [Fintype V], Fintype.card V = 5 * n →\n ∀ (G : SimpleGraph V), G.CliqueFree 3 →\n ∃ (H : SimpleGraph V),\n H ≤ G ∧ H.IsBipartite ∧ (G.edgeFinset \\ H.edgeFinset).card ≤ n^2",
"target": null... | [] | TheoremDB P9984 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9984",
"url": "https://theoremdb.org/statements/P9984",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9984",
"not... | {
"source_record": "P9984",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9984,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.316Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9ca9e10a-b1e9-447f-9c40-63d0e0625271","format_issues":null,"number":4714} |
1.0.0 | problem:theoremdb-p9986 | problem | theoremdb-p9986 | Finiteness of Three-Term Arithmetic Progressions of Consecutive Powerful Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $n$ is called powerful if for every prime $p$ dividing $n$, the square $p^2$ also divides $n$. Let $n_1 < n_2 < n_3 < \cdots$ denote the increasing sequence of all powerful numbers. For each positive integer $k$, consider the three-term set $\{n_k, n_{k+1}, n_{k+2}\}$ consisting of three consecutive ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9986",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9986",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_938 : answer(sorry) ↔ {P : Finset ℕ | (P : Set ℕ).IsAPOfLength 3 ∧ ∃ k,
P = {nth Powerful k, nth Powerful (k + 1), nth Powerful (k + 2)}}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_938 : answer(sorry) ↔ {P : Finset ℕ | (P : Set ℕ).IsAPOfLength 3 ∧ ∃ k,\n P = {nth Powerful k, nth Powerful (k + 1), nth Powerful (k + 2)}}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9986",
"format": "declaration-or... | [] | TheoremDB P9986 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9986",
"url": "https://theoremdb.org/statements/P9986",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9986",
"not... | {
"source_record": "P9986",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9986,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.117Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4458d8cd-f347-47cd-968b-785eeb56f2a2","format_issues":null,"number":5326} |
1.0.0 | problem:theoremdb-p9988 | problem | theoremdb-p9988 | Lehmer's totient problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Does there exist a composite number $ n > 1 $ such that Euler's totient function $ \varphi(n) $ divides $ n - 1 $? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9988",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9988",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lehmer_totient :
answer(sorry) ↔ ∃ n > 1, ¬Prime n ∧ Nat.totient n ∣ n - 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lehmer_totient :\n answer(sorry) ↔ ∃ n > 1, ¬Prime n ∧ Nat.totient n ∣ n - 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9988",
"format": "declaration-or-fragment",
"sha256": "c05d3f80a9b2b8ccf0f7d8bcaec8ed2ac09989baf0180a981... | [] | TheoremDB P9988 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9988",
"url": "https://theoremdb.org/statements/P9988",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9988",
"not... | {
"source_record": "P9988",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9988,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.174Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e13fed56-e57e-4355-bf2d-170bff3f0802","format_issues":null,"number":6039} |
1.0.0 | problem:theoremdb-p9990 | problem | theoremdb-p9990 | Erdős Problem 279 on Prime-Based Representations with Large Coefficients | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $k \geq 3$, a function $a$ assigning to every prime $p$ a residue class $a(p) \in \{0, 1, \ldots, p-1\}$ is called a choice of congruence classes. Determine whether the following statement holds: for every $k \geq 3$, there exists a choice of congruence classes $a$ and a positive integer $N$ such that ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9990",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9990",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_279 : answer(sorry) ↔ ∀ k : Nat, k ≥ 3 →
∃ a : Nat → Nat, ∃ N : Nat, (∀ p : Nat, p.Prime → a p < p) ∧
∀ n ≥ N, ∃ p : Nat, ∃ t ≥ k, p.Prime ∧ n = a p + t * p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_279 : answer(sorry) ↔ ∀ k : Nat, k ≥ 3 →\n ∃ a : Nat → Nat, ∃ N : Nat, (∀ p : Nat, p.Prime → a p < p) ∧\n ∀ n ≥ N, ∃ p : Nat, ∃ t ≥ k, p.Prime ∧ n = a p + t * p",
"target": null,
"source_url": "https://theoremdb.org/statements/P9990",
"format... | [] | TheoremDB P9990 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9990",
"url": "https://theoremdb.org/statements/P9990",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9990",
"not... | {
"source_record": "P9990",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:15+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9990,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:55.287Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c465cef8-2668-414d-99c2-ebe4371f4504","format_issues":null,"number":4487} |
1.0.0 | problem:theoremdb-p9992 | problem | theoremdb-p9992 | The 3-uniform partition relation at the first uncountable ordinal | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "community-map",
"evidence": "Inferred from community field: logic-and-foundations"
}
] | Let $\mathfrak{c}$ denote the cardinality of the continuum, and let $\omega_1 = \aleph_1$ denote the first uncountable ordinal. For each integer $n \geq 2$, consider the 3-uniform ordinal Ramsey property $\mathfrak{c} \to (\omega_1, n)^3_2$, defined as follows. For any 2-coloring of all 3-element subsets of the ordinal... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9992",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9992",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem omega_one :
answer(sorry) ↔
∀ᵉ (n : ℕ) (_ : 2 ≤ n),
OrdinalCardinalRamsey3 (𝔠).ord (Cardinal.aleph 1).ord n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem omega_one :\n answer(sorry) ↔\n ∀ᵉ (n : ℕ) (_ : 2 ≤ n),\n OrdinalCardinalRamsey3 (𝔠).ord (Cardinal.aleph 1).ord n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9992",
"format": "declaration-or-fragment",
"sha256": "0df... | [] | TheoremDB P9992 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9992",
"url": "https://theoremdb.org/statements/P9992",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9992",
"not... | {
"source_record": "P9992",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9992,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.340Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"74751ebf-9a56-4242-b97a-df222632d28c","format_issues":null,"number":6722} |
1.0.0 | problem:theoremdb-p9994 | problem | theoremdb-p9994 | Powerful Values of Factorial Plus One | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $m$ is called powerful if for every prime $p$ dividing $m$, the square $p^2$ also divides $m$. Equivalently, $m$ is powerful if and only if it can be written as $m = a^2 b^3$ for positive integers $a$ and $b$. For a sequence $a : \mathbb{N} \to \mathbb{N}$, we say that $a$ is eventually not powerful ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9994",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9994",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_936.variants.factorial_add_one :
answer(sorry) ↔ EventuallyNotPowerful (·! + 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_936.variants.factorial_add_one :\n answer(sorry) ↔ EventuallyNotPowerful (·! + 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9994",
"format": "declaration-or-fragment",
"sha256": "668691a0dacc41de61b645d433209601da6e362... | [] | TheoremDB P9994 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9994",
"url": "https://theoremdb.org/statements/P9994",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9994",
"not... | {
"source_record": "P9994",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9994,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.022Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"56eca34f-4fcd-4e43-ab11-d6f29236968e","format_issues":null,"number":6436} |
1.0.0 | problem:theoremdb-p9996 | problem | theoremdb-p9996 | Existence of the Mean Power Limit for Squarefree Gaps | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
$$
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alp... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9996",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9996",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_145 :
answer(sorry) ↔ ∀ α ≥ (0 : ℝ), ∃ β : ℝ,
atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_145 :\n answer(sorry) ↔ ∀ α ≥ (0 : ℝ), ∃ β : ℝ,\n atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9996",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9996 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9996",
"url": "https://theoremdb.org/statements/P9996",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9996",
"not... | {
"source_record": "P9996",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9996,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.852Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1c34a116-7c3c-4c89-a6a7-2e4df1fce552","format_issues":null,"number":5269} |
1.0.0 | problem:theoremdb-p9998 | problem | theoremdb-p9998 | Non-existence of monochromatic quantum graphs for even N ≥ 6 and D ≥ 3 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be an even integer with $N \geq 6$ and let $D \geq 3$ be an integer. A \textit{quantum graph} with $N$ vertices and $D$ colours over a coefficient domain is specified by a weight function assigning a value to each edge-coloured pair of vertices. For each assignment of colours to the $N$ vertices, one defines th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9998",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9998",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem_no_solution_ge6_ge3 :
answer(sorry) ↔
∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →
¬ ∃ W : WeightsN N D ℂ, EqSystemN N D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem_no_solution_ge6_ge3 :\n answer(sorry) ↔\n ∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →\n ¬ ∃ W : WeightsN N D ℂ, EqSystemN N D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9998",
"format": "declaration-or-fragment"... | [] | TheoremDB P9998 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9998",
"url": "https://theoremdb.org/statements/P9998",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9998",
"not... | {
"source_record": "P9998",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:05:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9998,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.290Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1df9b232-edf7-4bbc-9b57-0711d8c3084e","format_issues":null,"number":6280} |
1.0.0 | problem:tianyuan-logic-1-1 | problem | tianyuan-logic-1-1 | Combining chains of random Turing degrees — Problem 1.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Combining chains of random Turing degrees — Problem 1.1. https://arxiv.org/html/2608.26628v2#S1.Thmproblem1 Not an author-supplied code."
}
] | Problem 1.1 (Levin, private communications).
Are the following true?
-
Can every chain of random reals in the Turing degrees
be combined into a single random real?
-
Can every Turing ideal generated by random reals be
generated by the columns of a single random real? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Combining chains of random Turing degrees — Problem 1.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 1. George Barmpalias
The following problems concern Turing ideals generated by sufficiently
random reals.
Notation 1.1.
If $\alpha\in 2^{\omega}$, let $\alpha^{(i)}$ denote the real obtained from
$\alpha$ by deleting the digits $\alpha(k)$ at positions $k$ that are
divisible by $2^{i}$.
-
A chain is a sequence $(... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 1. George Barmpalias | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Combining chains of random Turing degrees — ... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:13.569Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"379c85c0-e6f4-4d8e-b993-1d29214be466","format_issues":null,"number":11816} |
1.0.0 | problem:tianyuan-logic-1-2 | problem | tianyuan-logic-1-2 | Full-measure sets and random degree chains — Problem 1.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Full-measure sets and random degree chains — Problem 1.2. https://arxiv.org/html/2608.26628v2#S1.Thmproblem2 Not an author-supplied code."
}
] | Problem 1.2 (Levin).
Is it true that, for every $X\subseteq 2^{\omega}$ with $\mu(X)=1$,
there exists $Y\subseteq 2^{\omega}$ with $\mu(Y)=1$ such that, for
every chain $B\subseteq Y$, there is some $\alpha\in X$ satisfying
$[B]=[\alpha]$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Full-measure sets and random degree chains — Problem 1.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 1. George Barmpalias
The following problems concern Turing ideals generated by sufficiently
random reals.
Notation 1.1.
If $\alpha\in 2^{\omega}$, let $\alpha^{(i)}$ denote the real obtained from
$\alpha$ by deleting the digits $\alpha(k)$ at positions $k$ that are
divisible by $2^{i}$.
-
A chain is a sequence $(... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 1. George Barmpalias | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Full-measure sets and random degree chains —... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem2",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:18.682Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9dab2b18-f8d6-46f4-8b4a-b232ab4c1f78","format_issues":null,"number":11817} |
1.0.0 | problem:tianyuan-logic-1-3 | problem | tianyuan-logic-1-3 | Chains of 2-random reals without a single generator — Problem 1.3 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Chains of 2-random reals without a single generator — Problem 1.3. https://arxiv.org/html/2608.26628v2#S1.Thmproblem3 Not an author-supplied code."
}
] | Problem 1.3.
Is there a chain
$B\subseteq\mathrm{ML}(\emptyset^{\prime})$
such that
$[B]\neq[\alpha]$
for every $\alpha\in\mathrm{ML}(\emptyset^{\prime})$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem3",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Chains of 2-random reals without a single generator — Problem 1.3. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 1. George Barmpalias
The following problems concern Turing ideals generated by sufficiently
random reals.
Notation 1.1.
If $\alpha\in 2^{\omega}$, let $\alpha^{(i)}$ denote the real obtained from
$\alpha$ by deleting the digits $\alpha(k)$ at positions $k$ that are
divisible by $2^{i}$.
-
A chain is a sequence $(... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 1. George Barmpalias | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Chains of 2-random reals without a single ge... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem3",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:18.804Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6d327303-2f03-4355-a226-56c0332522ef","format_issues":null,"number":11818} |
1.0.0 | problem:tianyuan-logic-1-4 | problem | tianyuan-logic-1-4 | Jumps and complexity within 2-random chains — Problem 1.4 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Jumps and complexity within 2-random chains — Problem 1.4. https://arxiv.org/html/2608.26628v2#S1.Thmproblem4 Not an author-supplied code."
}
] | Problem 1.4.
Toward a negative answer to Levin’s Problem 1.2, are the following true?
-
Are there $2$-random reals $x<_{T}z$ such that
$$
x^{\prime}\equiv_{T}z^{\prime}?
$$
-
Are there $2$-random reals $x<_{T}z$ such that
$$
\left|K(x\upharpoonright n)-K(z\upharpoonright n)\right|=O(1)?
$$
-
Is there a Turi... | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem4",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Jumps and complexity within 2-random chains — Problem 1.4. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 1. George Barmpalias
The following problems concern Turing ideals generated by sufficiently
random reals.
Notation 1.1.
If $\alpha\in 2^{\omega}$, let $\alpha^{(i)}$ denote the real obtained from
$\alpha$ by deleting the digits $\alpha(k)$ at positions $k$ that are
divisible by $2^{i}$.
-
A chain is a sequence $(... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 1. George Barmpalias | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Jumps and complexity within 2-random chains ... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem4",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:18.872Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a1ab8408-378a-4c11-a52a-6af093cb075b","format_issues":null,"number":11819} |
1.0.0 | problem:tianyuan-logic-1-5 | problem | tianyuan-logic-1-5 | Injective randomness-preserving one-way functions — Problem 1.5 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Injective randomness-preserving one-way functions — Problem 1.5. https://arxiv.org/html/2608.26628v2#S1.Thmproblem5 Not an author-supplied code."
}
] | Problem 1.5.
Is there an injective one-way function on the reals? More precisely,
is there a partial computable real function $f$ such that
-
$f$ is injective and preserves randomness,
-
the domain of $f$ has positive Lebesgue measure, and
-
$f(x)<_{T}x$
for a positive-measure set of reals $x$ in the domain o... | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem5",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Injective randomness-preserving one-way functions — Problem 1.5. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 1. George Barmpalias
The following problems concern Turing ideals generated by sufficiently
random reals.
Notation 1.1.
If $\alpha\in 2^{\omega}$, let $\alpha^{(i)}$ denote the real obtained from
$\alpha$ by deleting the digits $\alpha(k)$ at positions $k$ that are
divisible by $2^{i}$.
-
A chain is a sequence $(... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 1. George Barmpalias | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Injective randomness-preserving one-way func... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S1.Thmproblem5",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.073Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2c2211ef-9f74-477f-885f-3ce579ca04a8","format_issues":null,"number":11820} |
1.0.0 | problem:tianyuan-logic-10-1 | problem | tianyuan-logic-10-1 | Monotone induction and strong nesting — Problem 10.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Monotone induction and strong nesting — Problem 10.1. https://arxiv.org/html/2608.26628v2#S10.Thmproblem1 Not an author-supplied code."
}
] | Problem 10.1.
If $L_{\beta}\models{\mathit{V}=\mathit{HC}}$, then
$$
\left(\mathcal{P}(\omega)\cap L_{\beta}\models\boldsymbol{\Pi}_{m+1}^{1}\mathsf{-MI}\,\right)\iff\beta\mbox{ supports a strong }\Sigma_{m+1}\textrm{-nesting}.
$$ | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S10.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, D... | null | null | null | not-recorded | not-recorded | null | [] | [] | Monotone induction and strong nesting — Problem 10.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 10. Philip Welch
First we recall some basic notions in [[1](https://arxiv.org/html/2608.26628v2#bibg.bib1)]. Let $\beta$ be an ordinal. Suppose $M$ is a nonstandard model of $\mathsf{KP}$, and $\{\zeta_{i}:i\in\mathbb{N}\}$, $\{s_{i}:i\in\mathbb{N}\}$ are sequences of $M$-ordinals. If $\left(M,\{\zeta_{i}:i\in\mathbb{... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 10. Philip Welch | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Monotone induction and strong nesting — Prob... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S10.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does ... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.892Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c780d762-fe95-4ce1-9f36-56f59e8dfe67","format_issues":null,"number":11836} |
1.0.0 | problem:tianyuan-logic-11-1 | problem | tianyuan-logic-11-1 | A first-order theory with intermediate isomorphism complexity — Problem 11.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of A first-order theory with intermediate isomorphism complexity — Problem 11.1. https://arxiv.org/html/2608.26628v2#S11.Thmproblem1 Not an author-supplied code."
}
] | Problem 11.1.
Does there exist a complete first-order theory $T$ in a countable language such that $=\ <_{B}\ \cong\restriction_{\mathrm{Mod}(T)}\ <_{B}\ =^{+}$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S11.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, D... | null | null | null | not-recorded | not-recorded | null | [] | [] | A first-order theory with intermediate isomorphism complexity — Problem 11.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 11. Su Gao
This question is about equivalence relations given by complete first-order theories. The question has been circulating in the community since early 2000s.
Consider a complete first order theory $T$ in a countable language. Let
$\mathrm{Mod}(T)$ be the Polish space of countable models of $T$ with universe $... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 11. Su Gao | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; A first-order theory with intermediate isomo... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S11.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does ... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:21.015Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bbf02895-4059-45e9-8577-2cdd09b0649b","format_issues":null,"number":11837} |
1.0.0 | problem:tianyuan-logic-2-1 | problem | tianyuan-logic-2-1 | Outer automorphisms of oligomorphic groups — Problem 2.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Outer automorphisms of oligomorphic groups — Problem 2.1. https://arxiv.org/html/2608.26628v2#S2.Thmproblem1 Not an author-supplied code."
}
] | Problem 2.1.
Let $M$ be an $\omega$-categorical structure in a first order finite language. Is the group $\operatorname{Out}(G)$ finite? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Outer automorphisms of oligomorphic groups — Problem 2.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | Oligomorphic groups
Let $M$ be a countably infinite $\omega$-categorical structure in a
finite first-order signature, and let
$$
G=\operatorname{Aut}(M),
$$
equipped with the topology of pointwise convergence. Let
$\operatorname{Aut}(G)$ denote the group of continuous automorphisms
of the topological group $G$, and ... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | Oligomorphic groups | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Outer automorphisms of oligomorphic groups —... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.140Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"992f3a03-2b2d-497d-8bae-a2d09e8220e3","format_issues":null,"number":11821} |
1.0.0 | problem:tianyuan-logic-2-2 | problem | tianyuan-logic-2-2 | Strong jump traceability versus LR reducibility — Problem 2.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Strong jump traceability versus LR reducibility — Problem 2.2. https://arxiv.org/html/2608.26628v2#S2.Thmproblem2 Not an author-supplied code."
}
] | Problem 2.2.
Does $\leq_{SJT}$ imply $\leq_{LR}$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Strong jump traceability versus LR reducibility — Problem 2.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | Computability Theory
Assume throughout that $A,B\subset\mathbb{N}$. One writes
$$
A\leq_{SJT}B
$$
if for every order function $h$ there exists a uniformly
$B$-c.e. trace $(T_{n})$ satisfying that
$$
|T_{n}|\leq h(n)
$$
and
$$
J^{A}(n)\downarrow\Longrightarrow J^{A}(n)\in T_{n}.
$$
Also recall that
$$
A\leq_{LR}... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | Computability Theory | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Strong jump traceability versus LR reducibil... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem2",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.262Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a8a4f470-652a-4058-b291-52d83092d14d","format_issues":null,"number":11822} |
1.0.0 | problem:tianyuan-logic-2-3 | problem | tianyuan-logic-2-3 | Recursive maximal almost disjoint families in a closed class — Problem 2.3 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Recursive maximal almost disjoint families in a closed class — Problem 2.3. https://arxiv.org/html/2608.26628v2#S2.Thmproblem3 Not an author-supplied code."
}
] | Problem 2.3.
Is there a $\Pi^{0}_{1}$ class $C$ such that $|C\cap{\rm Rec}|=\infty$ and $C\cap{\rm Rec}$ is maximally almost disjoint among the recursive sets? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem3",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Recursive maximal almost disjoint families in a closed class — Problem 2.3. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | Computability Theory
Assume throughout that $A,B\subset\mathbb{N}$. One writes
$$
A\leq_{SJT}B
$$
if for every order function $h$ there exists a uniformly
$B$-c.e. trace $(T_{n})$ satisfying that
$$
|T_{n}|\leq h(n)
$$
and
$$
J^{A}(n)\downarrow\Longrightarrow J^{A}(n)\in T_{n}.
$$
Also recall that
$$
A\leq_{LR}... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | Computability Theory | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Recursive maximal almost disjoint families i... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S2.Thmproblem3",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.322Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"691e87bc-3ee9-47db-bbbe-3cff7fb706f2","format_issues":null,"number":11823} |
1.0.0 | problem:tianyuan-logic-3-1 | problem | tianyuan-logic-3-1 | Fourth-level quantifier-pattern degrees — Problem 3.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Fourth-level quantifier-pattern degrees — Problem 3.1. https://arxiv.org/html/2608.26628v2#S3.Thmproblem1 Not an author-supplied code."
}
] | Problem 3.1.
Determine the equivalence classes of quantifier patterns at the
fourth level of the arithmetical hierarchy:
-
How many $\equiv_{m}$-equivalence classes of
$\Sigma^{0}_{4}$ quantifier patterns are there?
-
How many $\equiv_{m}$-equivalence classes of
$\Pi^{0}_{4}$ quantifier patterns are there?
-
... | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S3.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Fourth-level quantifier-pattern degrees — Problem 3.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 3. Takayuki Kihara
The following problems concern a realizability-theoretic refinement of
many-one and Wadge reducibility.
For sets $A,B\subseteq\omega^{\omega}$, define
$$
A\leq_{m}B
$$
if there is a computable function
$$
\theta\colon\omega^{\omega}\to\omega^{\omega}
$$
such that
$$
x\in A\quad\Longleftrightar... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 3. Takayuki Kihara | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Fourth-level quantifier-pattern degrees — Pr... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S3.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.469Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3d7f1684-77eb-4b64-bc51-81233eac9900","format_issues":null,"number":11824} |
1.0.0 | problem:tianyuan-logic-3-2 | problem | tianyuan-logic-3-2 | Deciding reducibility of quantifier patterns — Problem 3.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Deciding reducibility of quantifier patterns — Problem 3.2. https://arxiv.org/html/2608.26628v2#S3.Thmproblem2 Not an author-supplied code."
}
] | Problem 3.2.
Is there a decision procedure which, given quantifier patterns
$$
\overline{P},\overline{Q}\in\{\exists,\forall,\exists^{\infty},\forall^{\infty}\}^{<\omega},
$$
decides whether
$$
\langle\overline{P}\rangle\leq_{m}\langle\overline{Q}\rangle?
$$ | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S3.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Deciding reducibility of quantifier patterns — Problem 3.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 3. Takayuki Kihara
The following problems concern a realizability-theoretic refinement of
many-one and Wadge reducibility.
For sets $A,B\subseteq\omega^{\omega}$, define
$$
A\leq_{m}B
$$
if there is a computable function
$$
\theta\colon\omega^{\omega}\to\omega^{\omega}
$$
such that
$$
x\in A\quad\Longleftrightar... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 3. Takayuki Kihara | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Deciding reducibility of quantifier patterns... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S3.Thmproblem2",
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"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.558Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"90a6cd22-3b4e-4933-ae86-0eb8b745c0f5","format_issues":null,"number":11825} |
1.0.0 | problem:tianyuan-logic-4-1 | problem | tianyuan-logic-4-1 | Kakeya sets whose points compute the halting problem — Problem 4.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Kakeya sets whose points compute the halting problem — Problem 4.1. https://arxiv.org/html/2608.26628v2#S4.Thmproblem1 Not an author-supplied code."
}
] | Problem 4.1.
Is there a Kakeya set $A\subseteq\mathbb{R}^{2}$ such that
$$
x\oplus y\geq_{T}\emptyset^{\prime}
$$
for every $(x,y)\in A$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S4.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Kakeya sets whose points compute the halting problem — Problem 4.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 4. Liang Yu
A set $A\subseteq\mathbb{R}^{2}$ is called a Kakeya set if, for every
$\alpha\in\mathbb{R}$, there exists $C_{\alpha}\in\mathbb{R}$ such that
$$
y=\alpha x+C_{\alpha}\quad\Longrightarrow\quad(x,y)\in A
$$
for every $(x,y)\in\mathbb{R}^{2}$. Equivalently,
$$
\bigl\{(x,y)\in\mathbb{R}^{2}:y=\alpha x+C_{\a... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 4. Liang Yu | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Kakeya sets whose points compute the halting... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S4.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.706Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"93be3838-e087-41a8-ab0c-f40ea1ce6610","format_issues":null,"number":11826} |
1.0.0 | problem:tianyuan-logic-4-2 | problem | tianyuan-logic-4-2 | Minimal subsets of Borel Kakeya sets — Problem 4.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Minimal subsets of Borel Kakeya sets — Problem 4.2. https://arxiv.org/html/2608.26628v2#S4.Thmproblem2 Not an author-supplied code."
}
] | Problem 4.2.
Does every Borel Kakeya set $A$ contain a minimal Kakeya set
$B\subseteq A$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S4.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Minimal subsets of Borel Kakeya sets — Problem 4.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 4. Liang Yu
A set $A\subseteq\mathbb{R}^{2}$ is called a Kakeya set if, for every
$\alpha\in\mathbb{R}$, there exists $C_{\alpha}\in\mathbb{R}$ such that
$$
y=\alpha x+C_{\alpha}\quad\Longrightarrow\quad(x,y)\in A
$$
for every $(x,y)\in\mathbb{R}^{2}$. Equivalently,
$$
\bigl\{(x,y)\in\mathbb{R}^{2}:y=\alpha x+C_{\a... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 4. Liang Yu | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Minimal subsets of Borel Kakeya sets — Probl... | {
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"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.799Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c8a5b3b3-df73-4764-804a-b1cf8212995c","format_issues":null,"number":11827} |
1.0.0 | problem:tianyuan-logic-5-1 | problem | tianyuan-logic-5-1 | Borel two-point sets — Problem 5.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Borel two-point sets — Problem 5.1. https://arxiv.org/html/2608.26628v2#S5.Thmproblem1 Not an author-supplied code."
}
] | Problem 5.1.
Is there a Borel two-point set? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S5.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Borel two-point sets — Problem 5.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 5. Jialiang He
These notes concern two related definability questions in infinite combinatorics and Borel model theory.
Definition 5.1.
A set $X\subseteq\mathbb{R}^{2}$ is called a two-point set, if for every line
$L\subseteq\mathbb{R}^{2}$,
$$
|X\cap L|=2.
$$
Theorem 5.2 (Larman [[1](https://arxiv.org/html/2608.2... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 5. Jialiang He | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Borel two-point sets — Problem 5.1",
"ur... | {
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"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:19.918Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"079eaea6-9097-422d-8834-3fab651948e7","format_issues":null,"number":11828} |
1.0.0 | problem:tianyuan-logic-5-2 | problem | tianyuan-logic-5-2 | Algebraic closures of Borel fields — Problem 5.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Algebraic closures of Borel fields — Problem 5.2. https://arxiv.org/html/2608.26628v2#S5.Thmproblem2 Not an author-supplied code."
}
] | Problem 5.2 (Montalbán–Nies [[3](https://arxiv.org/html/2608.26628v2#bibc.bib3), Question 2.8]).
Does every Borel field have a Borel algebraically closed extension? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S5.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Algebraic closures of Borel fields — Problem 5.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 5. Jialiang He
These notes concern two related definability questions in infinite combinatorics and Borel model theory.
Definition 5.1.
A set $X\subseteq\mathbb{R}^{2}$ is called a two-point set, if for every line
$L\subseteq\mathbb{R}^{2}$,
$$
|X\cap L|=2.
$$
Theorem 5.2 (Larman [[1](https://arxiv.org/html/2608.2... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 5. Jialiang He | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Algebraic closures of Borel fields — Problem... | {
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"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.034Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0aca193d-daad-457c-8c00-e98d0f442996","format_issues":null,"number":11829} |
1.0.0 | problem:tianyuan-logic-6-1 | problem | tianyuan-logic-6-1 | Choice needed for bases over a fixed field — Problem 6.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Choice needed for bases over a fixed field — Problem 6.1. https://arxiv.org/html/2608.26628v2#S6.Thmproblem1 Not an author-supplied code."
}
] | Problem 6.1.
For vector space over fixed $\mathbb{K}$, is choose a basis require the full Axiom of Choice? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S6.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Choice needed for bases over a fixed field — Problem 6.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 6. Chieu-Minh Tran
It is about Morley’s categoricity in power theorem.
According to [[1](https://arxiv.org/html/2608.26628v2#bibd.bib1)], there is the following consequence.
Theorem 6.1.
For a countable theory $T$, if it is $\kappa$ categorial for some $\kappa>\aleph_{0}$, then it is $\lambda$ categorial for all unc... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 6. Chieu-Minh Tran | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Choice needed for bases over a fixed field —... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S6.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.153Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"41eb7705-c289-49ae-983b-d595bcdfd8b7","format_issues":null,"number":11830} |
1.0.0 | problem:tianyuan-logic-7-1 | problem | tianyuan-logic-7-1 | A Davies example provable in ZFC — Problem 7.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of A Davies example provable in ZFC — Problem 7.1. https://arxiv.org/html/2608.26628v2#S7.Thmproblem1 Not an author-supplied code."
}
] | Problem 7.1.
Is there a Davies example provably in ZFC? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S7.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | A Davies example provable in ZFC — Problem 7.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 7. Theodore Slaman
In 1954, [[1](https://arxiv.org/html/2608.26628v2#bibe.bib1)] shows that for $E\subseteq\mathbb{R}^{2}$ an analytic set with Hausdorff dimension 1. Then, for almost all $\theta$, the projection of $E$ onto a line with slope $\theta$ has Hausdorff dimension 1. Later in 1969, [[2](https://arxiv.org/ht... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 7. Theodore Slaman | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; A Davies example provable in ZFC — Problem 7... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S7.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.271Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a69a465a-da20-47e0-b21b-46957c258e13","format_issues":null,"number":11831} |
1.0.0 | problem:tianyuan-logic-8-1 | problem | tianyuan-logic-8-1 | Forcing models satisfying the genericity property — Problem 8.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Forcing models satisfying the genericity property — Problem 8.1. https://arxiv.org/html/2608.26628v2#S8.Thmproblem1 Not an author-supplied code."
}
] | Problem 8.1.
Find more models $M$ and forcing notions $\mathbb{P}\in M$ such that $(*)$ is true. | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Forcing models satisfying the genericity property — Problem 8.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 8. Hang Zhang
The following questions are motivated by the problem whether hyper-hyperfiniteness implies hyperfiniteness.
Let $M$ be a countable model for a sufficiently large
fragment of ZFC. Suppose that a partial order $P$ is in $M$. Define the space $Gen^{P}_{M}$ of $M$-generics for $P$ by
$$
Gen^{P}_{M}=\{G\sub... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 8. Hang Zhang | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Forcing models satisfying the genericity pro... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.389Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"de8af487-d685-4c9c-94bb-5847dda793a6","format_issues":null,"number":11832} |
1.0.0 | problem:tianyuan-logic-8-2 | problem | tianyuan-logic-8-2 | Hyperfiniteness of the forcing equivalence relation — Problem 8.2 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Hyperfiniteness of the forcing equivalence relation — Problem 8.2. https://arxiv.org/html/2608.26628v2#S8.Thmproblem2 Not an author-supplied code."
}
] | Problem 8.2 ([[1](https://arxiv.org/html/2608.26628v2#bibf.bib1), Question 5.2]).
Is $\thicksim$ hyperfinite? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem2",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Hyperfiniteness of the forcing equivalence relation — Problem 8.2. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 8. Hang Zhang
The following questions are motivated by the problem whether hyper-hyperfiniteness implies hyperfiniteness.
Let $M$ be a countable model for a sufficiently large
fragment of ZFC. Suppose that a partial order $P$ is in $M$. Define the space $Gen^{P}_{M}$ of $M$-generics for $P$ by
$$
Gen^{P}_{M}=\{G\sub... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 8. Hang Zhang | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Hyperfiniteness of the forcing equivalence r... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem2",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.449Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5acbb2c9-78e0-4ad5-828f-54573258815b","format_issues":null,"number":11833} |
1.0.0 | problem:tianyuan-logic-8-3 | problem | tianyuan-logic-8-3 | Large-cardinal strength of the genericity property — Problem 8.3 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Large-cardinal strength of the genericity property — Problem 8.3. https://arxiv.org/html/2608.26628v2#S8.Thmproblem3 Not an author-supplied code."
}
] | Problem 8.3.
Suppose there are a model $M$ and a forcing notion $\mathbb{P}\in M$ such that $(*)$ is true. Can we deduce that $M\models[\exists\ \text{a large cardinal}]$? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem3",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Large-cardinal strength of the genericity property — Problem 8.3. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 8. Hang Zhang
The following questions are motivated by the problem whether hyper-hyperfiniteness implies hyperfiniteness.
Let $M$ be a countable model for a sufficiently large
fragment of ZFC. Suppose that a partial order $P$ is in $M$. Define the space $Gen^{P}_{M}$ of $M$-generics for $P$ by
$$
Gen^{P}_{M}=\{G\sub... | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 8. Hang Zhang | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Large-cardinal strength of the genericity pr... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S8.Thmproblem3",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.595Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d3c05f2d-7285-482a-882d-4e04c06f90ac","format_issues":null,"number":11834} |
1.0.0 | problem:tianyuan-logic-9-1 | problem | tianyuan-logic-9-1 | Extending the random-degree result to LR degrees — Problem 9.1 | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "source-map",
"evidence": "Editorial MSC2020 mapping of Extending the random-degree result to LR degrees — Problem 9.1. https://arxiv.org/html/2608.26628v2#S9.Thmproblem1 Not an author-supplied code."
}
] | Problem 9.1.
Does this hold with a larger degree notion, for example, LR-degrees? | unassessed | {
"key": "tianyuan-logic",
"name": "Tianyuan Workshop — Open Problems in Mathematical Logic",
"type": "preprint",
"url": "https://arxiv.org/html/2608.26628v2#S9.Thmproblem1",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Da... | null | null | null | not-recorded | not-recorded | null | [] | [] | Extending the random-degree result to LR degrees — Problem 9.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | 9. Daniel Turetsky
If $\mathbf{a}<\mathbf{b}<\mathbf{c}$ are Turing degrees, and $\mathbf{a}$, $\mathbf{c}$ contain different Martin-Löf randoms, then must $\mathbf{b}$ contain one? The answer is no. | Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today. | null | published | sharp | hard | [
"research-2026",
"tianyuan-logic"
] | 9. Daniel Turetsky | 2,026 | null | null | [
{
"kind": "paper",
"citation": "Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Extending the random-degree result to LR deg... | {
"source_record": "https://arxiv.org/html/2608.26628v2#S9.Thmproblem1",
"imported_at": "2026-09-10",
"status_as_of": "2608.26628v2; current status unassessed",
"evidence_label": "Licensed source statement",
"note": "Current research status is unassessed. The dated source poses this question; inclusion does n... | null | [
"current-status-unassessed"
] | [] | {
"created_at": "2026-09-10T05:04:07.731Z",
"updated_at": "2026-09-10T05:04:07.731Z",
"published_at": "2026-09-10T05:04:20.717Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"337bec0a-b288-4341-bc32-d71c2eb79a6d","format_issues":null,"number":11835} |
1.0.0 | problem:topp-10-simple-triangulation | problem | topp-10-simple-triangulation | A simpler deterministic linear-time polygon triangulation | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Find a deterministic algorithm triangulating every simple $n$-vertex polygon in $O(n)$ time with a substantially simpler construction and analysis than Chazelle's algorithm. Simplicity is an algorithm-design objective here, not a formally specified complexity measure. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p10",
"citation": "The Open Problems Project — Computational geometry, p10. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Find a deterministic algorithm triangulating every simple $n$-vertex polygon in $O(n)$ time with a substantially simpler construction and analysis than Chazelle's algorithm. Simplicity is an algorithm-design objective here, not a formally specified complexity measure. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | programme | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: A simpler deterministic linear-time polygon triangulation (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p10",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p10",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:37.818Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b8e84d9e-0136-4d9f-96fc-dc95f5a1e774","format_issues":null,"number":9525} |
1.0.0 | problem:topp-13-point-location-3d | problem | topp-13-point-location-3d | Linear-space logarithmic-time point location in three dimensions | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can a polyhedral subdivision of three-dimensional space with $n$ faces be represented using $O(n)$ space so that the cell containing a query point can be reported in $O(\log n)$ time? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p13",
"citation": "The Open Problems Project — Computational geometry, p13. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can a polyhedral subdivision of three-dimensional space with $n$ faces be represented using $O(n)$ space so that the cell containing a query point can be reported in $O(\log n)$ time? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Linear-space logarithmic-time point location in three dimensions (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p13",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p13",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:37.939Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fe061d1e-6b8f-46cd-b8e4-08d37f8d7510","format_issues":null,"number":9526} |
1.0.0 | problem:topp-15-output-sensitive-hull | problem | topp-15-output-sensitive-hull | Optimal output-sensitive convex hulls in higher dimensions | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For fixed $d$, determine the optimal time to construct the convex hull of $n$ points in $\mathbb{R}^d$, expressed in terms of $n$ and the number $f$ of output facets. Can the lower-bound target $O(n\log f+f)$ be attained throughout the output-size range? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p15",
"citation": "The Open Problems Project — Computational geometry, p15. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For fixed $d$, determine the optimal time to construct the convex hull of $n$ points in $\mathbb{R}^d$, expressed in terms of $n$ and the number $f$ of output facets. Can the lower-bound target $O(n\log f+f)$ be attained throughout the output-size range? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Optimal output-sensitive convex hulls in higher dimensions (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p15",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p15",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.000Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"518b2227-0a24-4856-ac3d-8ded852b8d26","format_issues":null,"number":9527} |
1.0.0 | problem:topp-16-count-polygonalizations | problem | topp-16-count-polygonalizations | Counting simple polygonalizations in polynomial time | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Given $n$ points in the plane, can one count their simple polygonalizations in polynomial time? A polygonalization is a non-self-intersecting polygon using exactly the given points as vertices; cyclic shifts and reversal describe the same polygon. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p16",
"citation": "The Open Problems Project — Computational geometry, p16. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Given $n$ points in the plane, can one count their simple polygonalizations in polynomial time? A polygonalization is a non-self-intersecting polygon using exactly the given points as vertices; cyclic shifts and reversal describe the same polygon. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Counting simple polygonalizations in polynomial time (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p16",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p16",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.258Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6b283dc7-7fc2-4701-ad7c-07811bfe8a87","format_issues":null,"number":9528} |
1.0.0 | problem:topp-22-minimum-link-path | problem | topp-22-minimum-link-path | Subquadratic exact minimum-link paths among planar obstacles | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For two points in the free space among polygonal obstacles with $n$ total vertices, can a connecting polygonal path with the fewest segments be found in $o(n^2)$ time? Segment orientations are unrestricted; an approximation to the minimum number of links does not settle the question. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p22",
"citation": "The Open Problems Project — Computational geometry, p22. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For two points in the free space among polygonal obstacles with $n$ total vertices, can a connecting polygonal path with the fewest segments be found in $o(n^2)$ time? Segment orientations are unrestricted; an approximation to the minimum number of links does not settle the question. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Subquadratic exact minimum-link paths among planar obstacles (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p22",
"note": null,
"sort_order": 0
},
{
"kind": "paper",
"citation": "Mitc... | {
"source_record": "https://topp.openproblem.net/p22",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.379Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"80a49e2d-7dd6-4246-8424-34028819c15a","format_issues":null,"number":9529} |
1.0.0 | problem:topp-23-vertex-floodlights | problem | topp-23-vertex-floodlights | Worst-case number of inward-facing vertex floodlights | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the smallest worst-case number of lights needed to illuminate a simple $n$-vertex polygon when each light has aperture $\pi$, is placed at a distinct polygon vertex, and faces inward: its illuminated half-plane lies locally inside the polygon at that vertex. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p23",
"citation": "The Open Problems Project — Computational geometry, p23. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the smallest worst-case number of lights needed to illuminate a simple $n$-vertex polygon when each light has aperture $\pi$, is placed at a distinct polygon vertex, and faces inward: its illuminated half-plane lies locally inside the polygon at that vertex. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Worst-case number of inward-facing vertex floodlights (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p23",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p23",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.525Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5abfebb6-12d3-4d16-a7f5-418e9572abb7","format_issues":null,"number":9530} |
1.0.0 | problem:topp-28-flip-connectivity | problem | topp-28-flip-connectivity | Connectivity of three-dimensional flip graphs | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For any finite point set in $\mathbb R^3$ with no four coplanar, can every tetrahedralization using all the points be transformed into any other using only geometric 2-to-3 and 3-to-2 bistellar flips? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p28",
"citation": "The Open Problems Project — Computational geometry, p28. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For any finite point set in $\mathbb R^3$ with no four coplanar, can every tetrahedralization using all the points be transformed into any other using only geometric 2-to-3 and 3-to-2 bistellar flips? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Connectivity of three-dimensional flip graphs (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p28",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p28",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:50.421Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9401f6b8-b9e7-4b65-8872-62448e0b1473","format_issues":null,"number":9520} |
1.0.0 | problem:topp-29-hamiltonian-tetrahedralizations | problem | topp-29-hamiltonian-tetrahedralizations | Hamiltonian tetrahedralizations of convex three-polytopes | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Does every convex polytope in $\mathbb{R}^3$ admit a tetrahedralization using its vertices and no Steiner points whose dual graph has a Hamiltonian path? Dual vertices represent tetrahedra and dual edges represent shared triangular faces. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p29",
"citation": "The Open Problems Project — Computational geometry, p29. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Does every convex polytope in $\mathbb{R}^3$ admit a tetrahedralization using its vertices and no Steiner points whose dual graph has a Hamiltonian path? Dual vertices represent tetrahedra and dual edges represent shared triangular faces. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Hamiltonian tetrahedralizations of convex three-polytopes (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p29",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p29",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.643Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45161eb5-379b-4394-b6b9-f257d5c05c48","format_issues":null,"number":9531} |
1.0.0 | problem:topp-3-line-voronoi-3d | problem | topp-3-line-voronoi-3d | Complexity of Voronoi diagrams of lines in three dimensions | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the worst-case combinatorial complexity of the Euclidean Voronoi diagram of $n$ lines in $\mathbb{R}^3$. In particular, can its complexity be bounded by $O(n^{2+\varepsilon})$ for every fixed $\varepsilon>0$? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p3",
"citation": "The Open Problems Project — Computational geometry, p3. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with attri... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the worst-case combinatorial complexity of the Euclidean Voronoi diagram of $n$ lines in $\mathbb{R}^3$. In particular, can its complexity be bounded by $O(n^{2+\varepsilon})$ for every fixed $\varepsilon>0$? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Complexity of Voronoi diagrams of lines in three dimensions (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p3",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p3",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult t... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:34.460Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d9f1a1e8-b539-4df7-b3e1-e452b3b76e66","format_issues":null,"number":9522} |
1.0.0 | problem:topp-31-segment-mirror-trap | problem | topp-31-segment-mirror-trap | Trapping every ray from a point with disjoint segment mirrors | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can finitely many two-sided straight mirror segments in the plane trap every ray emitted by a point away from the mirrors, so that no ray exits their convex hull? Require the closed segments to be pairwise disjoint, but treat endpoints as nonreflecting. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p31",
"citation": "The Open Problems Project — Computational geometry, p31. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can finitely many two-sided straight mirror segments in the plane trap every ray emitted by a point away from the mirrors, so that no ray exits their convex hull? Require the closed segments to be pairwise disjoint, but treat endpoints as nonreflecting. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Trapping every ray from a point with disjoint segment mirrors (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p31",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p31",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.790Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"44de87e3-9ba2-408d-b45e-58a564eb56b8","format_issues":null,"number":9532} |
1.0.0 | problem:topp-33-radical-separation | problem | topp-33-radical-separation | Separation bounds for sums of square roots | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For $1\leq a_i,b_i\leq n$, determine sharp bounds on the smallest positive $|\sum_{i=1}^{k}\sqrt{a_i}-\sum_{i=1}^{k}\sqrt{b_i}|$ as a function of $n,k$. Is the logarithm of its reciprocal polynomially bounded in $k$ and $\log n$? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p33",
"citation": "The Open Problems Project — Computational geometry, p33. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For $1\leq a_i,b_i\leq n$, determine sharp bounds on the smallest positive $|\sum_{i=1}^{k}\sqrt{a_i}-\sum_{i=1}^{k}\sqrt{b_i}|$ as a function of $n,k$. Is the logarithm of its reciprocal polynomially bounded in $k$ and $\log n$? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Separation bounds for sums of square roots (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p33",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p33",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:50.459Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"87646a84-4384-43bf-9de0-e1f134258ec0","format_issues":null,"number":9521} |
1.0.0 | problem:topp-34-extend-pseudosegments | problem | topp-34-extend-pseudosegments | Subdivision cost for extending pseudosegment arrangements | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | An arrangement of $n$ pseudosegments consists of finite planar curves meeting pairwise at most once. In the worst case, how many crossing points must be made into subdivision vertices so that the resulting pieces extend to doubly infinite pseudolines that still meet pairwise at most once? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p34",
"citation": "The Open Problems Project — Computational geometry, p34. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | An arrangement of $n$ pseudosegments consists of finite planar curves meeting pairwise at most once. In the worst case, how many crossing points must be made into subdivision vertices so that the resulting pieces extend to doubly infinite pseudolines that still meet pairwise at most once? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Subdivision cost for extending pseudosegment arrangements (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p34",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p34",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.802Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1f969e42-3542-46fd-b975-634f354d54bf","format_issues":null,"number":9533} |
1.0.0 | problem:topp-37-count-polyominoes | problem | topp-37-count-polyominoes | Enumeration and growth of polyominoes | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the number of edge-connected unions of $n$ unit lattice squares, or its asymptotic growth as $n\to\infty$. Distinguish counts modulo translations alone, modulo translations and rotations, and modulo translations, rotations and reflections; these are different enumeration sequences. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p37",
"citation": "The Open Problems Project — Computational geometry, p37. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the number of edge-connected unions of $n$ unit lattice squares, or its asymptotic growth as $n\to\infty$. Distinguish counts modulo translations alone, modulo translations and rotations, and modulo translations, rotations and reflections; these are different enumeration sequences. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Enumeration and growth of polyominoes (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p37",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p37",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.869Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1be9a9ec-95df-4215-95fc-1907fab43c6e","format_issues":null,"number":9534} |
1.0.0 | problem:topp-38-compatible-triangulations | problem | topp-38-compatible-triangulations | Compatible triangulations of two planar point sets | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Do any two general-position planar point sets with the same cardinality and the same number of convex-hull vertices have triangulations with isomorphic face lattices? The bijection between the two vertex sets may be chosen as part of the solution; no Steiner points are allowed. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p38",
"citation": "The Open Problems Project — Computational geometry, p38. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Do any two general-position planar point sets with the same cardinality and the same number of convex-hull vertices have triangulations with isomorphic face lattices? The bijection between the two vertex sets may be chosen as part of the solution; no Steiner points are allowed. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Compatible triangulations of two planar point sets (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p38",
"note": null,
"sort_order": 0
},
{
"kind": "paper",
"citation": "Barnson et al.... | {
"source_record": "https://topp.openproblem.net/p38",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:38.957Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1221da89-1d12-4d3b-8054-71e718ace771","format_issues":null,"number":9535} |
1.0.0 | problem:topp-40-count-pointed-pseudotriangulations | problem | topp-40-count-pointed-pseudotriangulations | Comparing counts of triangulations and pointed pseudotriangulations | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For a planar point set, is the number of pointed pseudotriangulations at least the number of triangulations, with equality only for convex position? A pseudotriangulation partitions its convex hull into polygons with three convex corners; pointedness requires an incident angle greater than a straight angle at every ver... | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p40",
"citation": "The Open Problems Project — Computational geometry, p40. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For a planar point set, is the number of pointed pseudotriangulations at least the number of triangulations, with equality only for convex position? A pseudotriangulation partitions its convex hull into polygons with three convex corners; pointedness requires an incident angle greater than a straight angle at every ver... | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Comparing counts of triangulations and pointed pseudotriangulations (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p40",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p40",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.049Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c329507d-4293-4918-b544-0ca85e59e8e4","format_issues":null,"number":9536} |
1.0.0 | problem:topp-41-sort-pairwise-sums | problem | topp-41-sort-pairwise-sums | Optimal time to sort all pairwise sums | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Given two lists $X$ and $Y$ of $n$ real numbers each, determine the optimal running time for outputting the $n^2$ sums $x+y$ in sorted order. Can the additive structure yield an $O(n^2)$-time algorithm in the standard real-arithmetic model? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p41",
"citation": "The Open Problems Project — Computational geometry, p41. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Given two lists $X$ and $Y$ of $n$ real numbers each, determine the optimal running time for outputting the $n^2$ sums $x+y$ in sorted order. Can the additive structure yield an $O(n^2)$-time algorithm in the standard real-arithmetic model? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Optimal time to sort all pairwise sums (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p41",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p41",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.057Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f9fc465e-a375-4c0a-8b4f-4b31f6ad9379","format_issues":null,"number":9537} |
1.0.0 | problem:topp-42-vertex-unfolding | problem | topp-42-vertex-unfolding | Vertex unfoldings of polyhedra with nonsimplicial faces | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Does every closed polyhedron with simply connected faces admit a nonoverlapping planar unfolding obtained by cutting edges while retaining enough vertex contacts to keep all faces in one connected piece? Faces must remain intact; the unrestricted question includes nonsimplicial polyhedra. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p42",
"citation": "The Open Problems Project — Computational geometry, p42. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Does every closed polyhedron with simply connected faces admit a nonoverlapping planar unfolding obtained by cutting edges while retaining enough vertex contacts to keep all faces in one connected piece? Faces must remain intact; the unrestricted question includes nonsimplicial polyhedra. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Vertex unfoldings of polyhedra with nonsimplicial faces (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p42",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p42",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.202Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"32f5864c-029f-4c5c-9c37-69f0cb267592","format_issues":null,"number":9538} |
1.0.0 | problem:topp-43-nonconvex-general-unfolding | problem | topp-43-nonconvex-general-unfolding | General unfoldings of nonconvex polyhedra | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can the surface of every closed nonconvex polyhedron be cut into a single connected piece that flattens into the plane without overlap? Cuts may cross face interiors, so they need not follow the original edges. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p43",
"citation": "The Open Problems Project — Computational geometry, p43. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can the surface of every closed nonconvex polyhedron be cut into a single connected piece that flattens into the plane without overlap? Cuts may cross face interiors, so they need not follow the original edges. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: General unfoldings of nonconvex polyhedra (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p43",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p43",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.350Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1130c39d-149f-418a-a6fb-c15e27a3051f","format_issues":null,"number":9539} |
1.0.0 | problem:topp-45-universal-point-sets | problem | topp-45-universal-point-sets | Smallest universal point sets for planar graph drawings | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the minimum size $u(n)$ of a planar point set on which every $n$-vertex planar graph has a crossing-free straight-line drawing, using a subset of the points as its vertices. In particular, is $u(n)=O(n)$ possible? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p45",
"citation": "The Open Problems Project — Computational geometry, p45. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the minimum size $u(n)$ of a planar point set on which every $n$-vertex planar graph has a crossing-free straight-line drawing, using a subset of the points as its vertices. In particular, is $u(n)=O(n)$ possible? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Smallest universal point sets for planar graph drawings (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p45",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p45",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.358Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8f454514-478e-406d-a3e7-2c0c5c21f230","format_issues":null,"number":9540} |
1.0.0 | problem:topp-46-two-bend-3d-drawings | problem | topp-46-two-bend-3d-drawings | Two-bend orthogonal drawings of degree-six graphs | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Does every finite simple graph of maximum degree at most six admit a drawing with vertices on $\mathbb{Z}^3$ and edges following axis-parallel lattice paths with at most two bends each? Distinct paths may meet only at a shared graph endpoint. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p46",
"citation": "The Open Problems Project — Computational geometry, p46. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Does every finite simple graph of maximum degree at most six admit a drawing with vertices on $\mathbb{Z}^3$ and edges following axis-parallel lattice paths with at most two bends each? Distinct paths may meet only at a shared graph endpoint. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Two-bend orthogonal drawings of degree-six graphs (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p46",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p46",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.449Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1db9a808-3220-4974-84e4-09df2d542b19","format_issues":null,"number":9541} |
1.0.0 | problem:topp-49-maximum-euclidean-tsp | problem | topp-49-maximum-euclidean-tsp | Complexity of the planar Euclidean maximum traveling salesman problem | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the computational complexity of finding a Hamiltonian tour of maximum total Euclidean length through a given finite set of planar points. The metric is ordinary Euclidean distance, and the objective is to maximize the sum of tour-edge lengths. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p49",
"citation": "The Open Problems Project — Computational geometry, p49. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the computational complexity of finding a Hamiltonian tour of maximum total Euclidean length through a given finite set of planar points. The metric is ordinary Euclidean distance, and the objective is to maximize the sum of tour-edge lengths. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Complexity of the planar Euclidean maximum traveling salesman problem (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p49",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p49",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.508Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"60276f85-a3da-4021-b5fa-6438fa3c32ef","format_issues":null,"number":9542} |
1.0.0 | problem:topp-5-euclidean-mst | problem | topp-5-euclidean-mst | Near-linear Euclidean minimum spanning trees in fixed dimension | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For each fixed dimension $d\geq3$, can an exact Euclidean minimum spanning tree of $n$ points in $\mathbb{R}^d$ be computed in time close to $O(n\log n)$? The target concerns the geometric input, without explicitly constructing its complete weighted graph. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p5",
"citation": "The Open Problems Project — Computational geometry, p5. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with attri... | null | null | null | not-recorded | not-recorded | null | [] | [] | For each fixed dimension $d\geq3$, can an exact Euclidean minimum spanning tree of $n$ points in $\mathbb{R}^d$ be computed in time close to $O(n\log n)$? The target concerns the geometric input, without explicitly constructing its complete weighted graph. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Near-linear Euclidean minimum spanning trees in fixed dimension (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p5",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p5",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult t... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:37.578Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c1c8e8ba-192e-4616-be1b-951c2bc3eb39","format_issues":null,"number":9523} |
1.0.0 | problem:topp-54-solid-grid-tsp | problem | topp-54-solid-grid-tsp | Shortest traveling-salesman tours in solid grid graphs | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the complexity of a shortest closed walk visiting every vertex of a solid planar grid graph. Vertices are lattice points, all unit-distance edges are included, the graph has no holes, and lengths are measured by its graph distances; repeated vertices are allowed. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p54",
"citation": "The Open Problems Project — Computational geometry, p54. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the complexity of a shortest closed walk visiting every vertex of a solid planar grid graph. Vertices are lattice points, all unit-distance edges are included, the graph has no holes, and lengths are measured by its graph distances; repeated vertices are allowed. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Shortest traveling-salesman tours in solid grid graphs (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p54",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p54",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.626Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ae360e7d-959f-4c56-a2d1-90f7070d48a0","format_issues":null,"number":9543} |
1.0.0 | problem:topp-55-pallet-loading | problem | topp-55-pallet-loading | Complexity of rectangular pallet loading | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Given positive integer dimensions $A,B,a,b$ and an integer $n$ in binary, decide whether $n$ interior-disjoint $a\times b$ rectangles fit inside an $A\times B$ rectangle. Each small rectangle may use either axis-parallel orientation. Determine the complexity in terms of the compact input length. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p55",
"citation": "The Open Problems Project — Computational geometry, p55. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Given positive integer dimensions $A,B,a,b$ and an integer $n$ in binary, decide whether $n$ interior-disjoint $a\times b$ rectangles fit inside an $A\times B$ rectangle. Each small rectangle may use either axis-parallel orientation. Determine the complexity in terms of the compact input length. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Complexity of rectangular pallet loading (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p55",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p55",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.717Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d9209bb4-6800-437c-b455-433273c27e9b","format_issues":null,"number":9544} |
1.0.0 | problem:topp-58-avoid-monochromatic-triangles | problem | topp-58-avoid-monochromatic-triangles | Three-colorings avoiding a prescribed monochromatic triangle | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For each fixed nondegenerate planar triangle, does there exist a three-coloring of the entire plane with no congruent copy whose three vertices all have one color? The coloring may depend on the chosen triangle, and copies may occur in arbitrary positions and orientations. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p58",
"citation": "The Open Problems Project — Computational geometry, p58. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For each fixed nondegenerate planar triangle, does there exist a three-coloring of the entire plane with no congruent copy whose three vertices all have one color? The coloring may depend on the chosen triangle, and copies may occur in arbitrary positions and orientations. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Three-colorings avoiding a prescribed monochromatic triangle (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p58",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p58",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.750Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9b521e0b-f43c-4d1d-9f59-fcfa2b8a5c1d","format_issues":null,"number":9545} |
1.0.0 | problem:topp-59-circular-square-partitions | problem | topp-59-circular-square-partitions | Optimal circularity of convex partitions of a square | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For a convex piece, define circularity as its smallest enclosing-circle radius divided by its largest inscribed-circle radius. Determine the infimum, over finite convex polygonal partitions of a square, of the largest piece circularity. Is this infimum attained by any finite partition? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p59",
"citation": "The Open Problems Project — Computational geometry, p59. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For a convex piece, define circularity as its smallest enclosing-circle radius divided by its largest inscribed-circle radius. Determine the infimum, over finite convex polygonal partitions of a square, of the largest piece circularity. Is this infimum attained by any finite partition? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Optimal circularity of convex partitions of a square (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p59",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p59",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.815Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7095c0b9-9512-44f4-ba34-408d51a41426","format_issues":null,"number":9546} |
1.0.0 | problem:topp-6-euclidean-matching | problem | topp-6-euclidean-matching | Exact minimum-length matching of planar points | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Determine the optimal running time for finding a perfect matching of $2n$ planar points that minimizes the sum of Euclidean edge lengths. The task is exact optimization, with no bipartite restriction. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p6",
"citation": "The Open Problems Project — Computational geometry, p6. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with attri... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine the optimal running time for finding a perfect matching of $2n$ planar points that minimizes the sum of Euclidean edge lengths. The task is exact optimization, with no bipartite restriction. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Exact minimum-length matching of planar points (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p6",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p6",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult t... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:37.698Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"99e6b48a-3163-4f3d-a058-32e9727bf34b","format_issues":null,"number":9524} |
1.0.0 | problem:topp-62-volume-perimeter-halving | problem | topp-62-volume-perimeter-halving | Maximum volume from a perimeter-halving fold | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Among convex planar shapes of unit area, which yields the largest volume by perimeter-halving folding? Choose two boundary points that divide the perimeter equally and identify the two intervening boundary arcs by equal arc length to obtain the associated convex surface in three dimensions. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p62",
"citation": "The Open Problems Project — Computational geometry, p62. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Among convex planar shapes of unit area, which yields the largest volume by perimeter-halving folding? Choose two boundary points that divide the perimeter equally and identify the two intervening boundary arcs by equal arc length to obtain the associated convex surface in three dimensions. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Maximum volume from a perimeter-halving fold (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p62",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p62",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:39.932Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e018449d-5922-4040-9806-7e9a19017325","format_issues":null,"number":9547} |
1.0.0 | problem:topp-63-dynamic-nearest-neighbors | problem | topp-63-dynamic-nearest-neighbors | Logarithmic dynamic nearest-neighbor queries in the plane | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can a data structure maintain $n$ planar points under insertion and deletion, and return the Euclidean nearest neighbor of any query point, with $O(\log n)$ time for each operation? Specify whether update guarantees are amortized or worst-case. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p63",
"citation": "The Open Problems Project — Computational geometry, p63. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can a data structure maintain $n$ planar points under insertion and deletion, and return the Euclidean nearest neighbor of any query point, with $O(\log n)$ time for each operation? Specify whether update guarantees are amortized or worst-case. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Logarithmic dynamic nearest-neighbor queries in the plane (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p63",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p63",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.021Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e963f87b-6eb0-4dd4-bdc5-a0760e3fcfce","format_issues":null,"number":9548} |
1.0.0 | problem:topp-64-polycube-edge-unfolding | problem | topp-64-polycube-edge-unfolding | Edge unfoldings of genus-zero polycubes | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can every genus-zero polyhedron formed by joining unit cubes face to face be unfolded into one planar piece without overlap by cutting along exposed cube edges? All cube edges on the boundary are available, including those lying within a larger flat face. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p64",
"citation": "The Open Problems Project — Computational geometry, p64. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can every genus-zero polyhedron formed by joining unit cubes face to face be unfolded into one planar piece without overlap by cutting along exposed cube edges? All cube edges on the boundary are available, including those lying within a larger flat face. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Edge unfoldings of genus-zero polycubes (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p64",
"note": null,
"sort_order": 0
},
{
"kind": "paper",
"citation": "Edge-Unfolding Polycubes ... | {
"source_record": "https://topp.openproblem.net/p64",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.169Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9e305c65-7db2-4db1-b451-c6f8d31549b2","format_issues":null,"number":9549} |
1.0.0 | problem:topp-66-point-set-reflexivity | problem | topp-66-point-set-reflexivity | Worst-case reflexivity of planar point sets | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For a planar point set $S$ in general position, let $\rho(S)$ be the minimum number of reflex vertices in a simple polygon using exactly $S$ as its vertices. Determine $\rho(n)=\max_{|S|=n}\rho(S)$. Is $\rho(n)=\lfloor n/4\rfloor$? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p66",
"citation": "The Open Problems Project — Computational geometry, p66. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For a planar point set $S$ in general position, let $\rho(S)$ be the minimum number of reflex vertices in a simple polygon using exactly $S$ as its vertices. Determine $\rho(n)=\max_{|S|=n}\rho(S)$. Is $\rho(n)=\lfloor n/4\rfloor$? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Worst-case reflexivity of planar point sets (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p66",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p66",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.288Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f86f5efe-9bd7-46f2-abd8-19dd140821b0","format_issues":null,"number":9550} |
1.0.0 | problem:topp-68-die-labeled-board | problem | topp-68-die-labeled-board | Rolling a die over a fully labeled rectangular board | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | A rectangular grid board assigns a value from one to six to every cell. Determine the complexity of deciding whether a standard labeled die, from the specified corner and initial orientation, can roll across the board visiting each cell exactly once while its top face always matches the cell label. No free or blocked c... | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p68",
"citation": "The Open Problems Project — Computational geometry, p68. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | A rectangular grid board assigns a value from one to six to every cell. Determine the complexity of deciding whether a standard labeled die, from the specified corner and initial orientation, can roll across the board visiting each cell exactly once while its top face always matches the cell label. No free or blocked c... | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Rolling a die over a fully labeled rectangular board (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p68",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p68",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.385Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7e2c2336-3c48-4442-a575-d06069777c43","format_issues":null,"number":9551} |
1.0.0 | problem:topp-72-regular-pentagon-surfaces | problem | topp-72-regular-pentagon-surfaces | Spherical polyhedral surfaces made of regular pentagons | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Must every closed polyhedral surface homeomorphic to $S^2$, composed of congruent regular pentagons and immersed in $\mathbb{R}^3$, arise as the boundary of a union of solid regular dodecahedra joined along whole faces? | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p72",
"citation": "The Open Problems Project — Computational geometry, p72. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Must every closed polyhedral surface homeomorphic to $S^2$, composed of congruent regular pentagons and immersed in $\mathbb{R}^3$, arise as the boundary of a union of solid regular dodecahedra joined along whole faces? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Spherical polyhedral surfaces made of regular pentagons (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p72",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p72",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.418Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7128d0a0-42f4-4f21-8c4f-9f2842182469","format_issues":null,"number":9552} |
1.0.0 | problem:topp-75-geometric-edge-coloring | problem | topp-75-geometric-edge-coloring | Coloring geometric complete graphs into noncrossing classes | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For a general-position set of $n$ planar points, color every segment joining two points so that properly crossing segments have different colors. Determine the worst-case minimum number of colors as a function of $n$; shared endpoints do not count as proper crossings. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p75",
"citation": "The Open Problems Project — Computational geometry, p75. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For a general-position set of $n$ planar points, color every segment joining two points so that properly crossing segments have different colors. Determine the worst-case minimum number of colors as a function of $n$; shared endpoints do not count as proper crossings. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Coloring geometric complete graphs into noncrossing classes (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p75",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p75",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.453Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4aebb0d0-0c18-4e63-8c27-7562f253bb42","format_issues":null,"number":9553} |
1.0.0 | problem:topp-76-equiprojective-polyhedra | problem | topp-76-equiprojective-polyhedra | Constructive classification of equiprojective polyhedra | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | For each $k$, classify or construct all $k$-equiprojective polyhedra: polyhedra whose orthogonal projection is a $k$-gon in every direction not parallel to a face. The source already gives an edge-face-pair characterization; the remaining objective is an explicit classification and construction. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p76",
"citation": "The Open Problems Project — Computational geometry, p76. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | For each $k$, classify or construct all $k$-equiprojective polyhedra: polyhedra whose orthogonal projection is a $k$-gon in every direction not parallel to a face. The source already gives an edge-face-pair characterization; the remaining objective is an explicit classification and construction. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Constructive classification of equiprojective polyhedra (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p76",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p76",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.552Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"08d15ceb-070a-4558-8378-5c396e1d7eb0","format_issues":null,"number":9554} |
1.0.0 | problem:topp-77-zipper-unfolding | problem | topp-77-zipper-unfolding | Zipper unfoldings of convex polyhedra | {
"main": "computer science",
"sub": "computational geometry",
"slug": "computational-geometry",
"name": "Computational geometry",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68U05",
"basis": "community-map",
"evidence": "Inferred from community field: computational-geometry"
}
] | Can every convex polyhedron be flattened without overlap after cutting along one simple path through all its vertices? This zipper path may traverse face interiors, and is not required to follow the edge graph. | unassessed | {
"key": "topp",
"name": "The Open Problems Project — Computational geometry",
"type": "problem-list",
"url": "https://topp.openproblem.net/p77",
"citation": "The Open Problems Project — Computational geometry, p77. Inspected 2026-09-09.",
"doi": null,
"arxiv_id": null,
"license": "Academic use with att... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can every convex polyhedron be flattened without overlap after cutting along one simple path through all its vertices? This zipper path may traverse face interiors, and is not required to follow the edge graph. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature. | null | null | published | sharp | hard | [
"computational-geometry",
"research-list-selection",
"status-unassessed"
] | null | null | null | null | [
{
"kind": "website",
"citation": "The Open Problems Project — Computational geometry: Zipper unfoldings of convex polyhedra (inspected 2026-09-09)",
"url": "https://topp.openproblem.net/p77",
"note": null,
"sort_order": 0
}
] | {
"source_record": "https://topp.openproblem.net/p77",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Research-list selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult ... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T15:08:22.424Z",
"updated_at": "2026-09-09T15:08:22.424Z",
"published_at": "2026-09-09T15:08:40.674Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d2ef457a-ddaf-4334-a9d1-40b15222688d","format_issues":null,"number":9555} |
1.0.0 | problem:tsp-below-two-to-the-n | problem | tsp-below-two-to-the-n | Beating 2^n for the travelling salesman problem | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | Is there an algorithm solving the travelling salesman problem exactly on $n$ cities with arbitrary non-negative weights in time $O(c^n)$ for some constant $c < 2$? | open | {
"key": "community",
"name": "Community consensus",
"type": "journal",
"url": null,
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Original wording",
"notes": "Questions that are widely regarded as open but have no single canonical citation; written here from the surrounding literature.... | null | null | null | not-recorded | not-recorded | null | [] | [] | Held–Karp has stood since 1962. No exact algorithm is known with running time O(1.99^n). | The Held–Karp dynamic program runs in $O(2^n n^2)$ time and $O(2^n n)$ space. Six decades later the exponential base has not been improved in the general weighted case, making this one of the cleanest stagnation points in exact algorithms. | - For the *unweighted* Hamiltonicity problem, Björklund (2010) gave an $O(1.657^n)$ randomised algorithm for undirected graphs, so the barrier is specific to weights.
- Bounded-degree and bounded-treewidth cases admit faster algorithms.
- Under the Strong Exponential Time Hypothesis, some related covering problems prov... | null | published | sharp | hard | [
"dynamic-programming",
"exact-algorithms",
"exponential-time"
] | Held and Karp lineage | 1,962 | null | null | [] | {
"source_record": null,
"imported_at": null,
"status_as_of": null,
"evidence_label": null,
"note": null,
"has_full_text": null,
"original_file": null,
"sha256": null
} | null | [] | [] | {
"created_at": "2026-09-07T23:45:31.857Z",
"updated_at": "2026-09-07T23:45:31.857Z",
"published_at": "2026-09-07T23:45:35.152Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a7af6b73-7df5-4a36-945a-8ffe37b9c8cd","format_issues":null,"number":35} |
1.0.0 | problem:twin-prime-conjecture | problem | twin-prime-conjecture | The twin prime conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Prove or disprove: there are infinitely many primes $p$ for which $p+2$ is prime. The quantitative form, due to Hardy and Littlewood, predicts
$$\pi_2(x) \sim 2\Pi_2 \int_2^x \frac{dt}{(\log t)^2}, \qquad \Pi_2 = \prod_{p>2}\left(1 - \frac{1}{(p-1)^2}\right).$$ | open | {
"key": "community",
"name": "Community consensus",
"type": "problem-list",
"url": null,
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Original wording",
"notes": "Questions that are widely regarded as open but have no single canonical citation; written here from the surrounding litera... | import Std
-- Original Frontier Science preview transcription; not an upstream formalization.
-- A statement only: no proof or claim of verification.
def TwinPrimeConjecture : Prop :=
∀ N : Nat, ∃ p : Nat,
N < p ∧
(2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧
(2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → ... | null | null | not-recorded | not-recorded | ∀ N : Nat, ∃ p : Nat, N < p ∧ (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d = p) ∧ (2 ≤ p + 2 ∧ ∀ d : Nat, (p + 2) % d = 0 → d = 1 ∨ d = p + 2) | [
{
"language": "Lean 4",
"code": "import Std\n\n-- Original Frontier Science preview transcription; not an upstream formalization.\n-- A statement only: no proof or claim of verification.\ndef TwinPrimeConjecture : Prop :=\n ∀ N : Nat, ∃ p : Nat,\n N < p ∧\n (2 ≤ p ∧ ∀ d : Nat, p % d = 0 → d = 1 ∨ d =... | [] | Are there infinitely many primes p such that p + 2 is also prime? | null | - Zhang (2013) proved that $\liminf_n (p_{n+1}-p_n) < 7\times 10^7$, the first finite bound.
- Maynard and, independently, Tao gave a different sieve; the Polymath8 project reduced the bound to $246$ unconditionally, and to $6$ under the generalised Elliott–Halberstam conjecture.
- Chen (1973) proved there are infinite... | null | published | sharp | notorious | [
"prime-gaps",
"primes",
"sieve-theory"
] | Attributed to de Polignac (1849) | 1,849 | null | null | [
{
"kind": "paper",
"citation": "Y. Zhang, Bounded gaps between primes, Ann. of Math. 179 (2014), 1121–1174.",
"url": null,
"note": null,
"sort_order": 0
},
{
"kind": "paper",
"citation": "J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015), 383–413.",
"url": "https://... | {
"source_record": null,
"imported_at": null,
"status_as_of": null,
"evidence_label": null,
"note": null,
"has_full_text": null,
"original_file": null,
"sha256": null
} | null | [
"formalization-not-compiled-during-export"
] | [] | {
"created_at": "2026-09-07T23:45:31.857Z",
"updated_at": "2026-09-07T23:45:31.857Z",
"published_at": "2026-09-07T23:45:34.811Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e259931f-368a-4968-9963-7f38298da884","format_issues":null,"number":6} |
1.0.0 | problem:union-closed-sets-conjecture | problem | union-closed-sets-conjecture | Frankl's union-closed sets conjecture | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite family $\mathcal{F}$ of finite sets is *union-closed* if $A, B \in \mathcal{F}$ implies $A\cup B \in \mathcal{F}$. Conjecture: if $\mathcal{F}$ is union-closed and contains a non-empty set, then some element belongs to at least $|\mathcal{F}|/2$ of the members of $\mathcal{F}$. | partially-solved | {
"key": "open-problem-garden",
"name": "Open Problem Garden",
"type": "problem-list",
"url": "https://www.openproblemgarden.org/",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Community wiki",
"notes": "A long-running community wiki, strongest in graph theory and combinatorics."
} | null | null | null | not-recorded | not-recorded | null | [] | [] | In any finite union-closed family, must some element lie in at least half the sets? | Attributed to Péter Frankl in 1979. For four decades no constant $c>0$ was known for which some element appears in a $c$ fraction of the sets, despite the statement being accessible to an undergraduate. | - Gilmer (2022) broke the barrier with an entropy argument, obtaining $c \approx 0.01$.
- Within weeks several groups independently improved the constant to $(3-\sqrt5)/2 \approx 0.38$, which is the natural limit of Gilmer's method.
- The gap between $0.38$ and $0.5$ appears to require a genuinely different idea.
- The... | null | published | sharp | hard | [
"entropy-method",
"extremal-set-theory",
"frankl"
] | Péter Frankl | 1,979 | null | null | [
{
"kind": "paper",
"citation": "J. Gilmer, A constant lower bound for the union-closed sets conjecture (2022).",
"url": "https://arxiv.org/abs/2211.09055",
"note": null,
"sort_order": 0
}
] | {
"source_record": null,
"imported_at": null,
"status_as_of": null,
"evidence_label": null,
"note": null,
"has_full_text": null,
"original_file": null,
"sha256": null
} | null | [] | [] | {
"created_at": "2026-09-07T23:45:31.857Z",
"updated_at": "2026-09-07T23:45:31.857Z",
"published_at": "2026-09-07T23:45:34.912Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d1cdec7c-dfbc-4ccd-9ab3-0f4548652e06","format_issues":null,"number":9} |
1.0.0 | problem:unique-games-conjecture | problem | unique-games-conjecture | The Unique Games Conjecture | {
"main": "computer science",
"sub": "computational complexity",
"slug": "computational-complexity",
"name": "Computational complexity",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68Q",
"basis": "community-map",
"evidence": "Inferred from community field: computational-complexity"
}
] | For every $\varepsilon, \delta > 0$ there is $k$ such that it is NP-hard, given a unique games instance with label set of size $k$, to distinguish instances where at least a $1-\varepsilon$ fraction of constraints can be satisfied from instances where at most a $\delta$ fraction can be. | open | {
"key": "community",
"name": "Community consensus",
"type": "journal",
"url": null,
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Original wording",
"notes": "Questions that are widely regarded as open but have no single canonical citation; written here from the surrounding literature.... | null | null | null | not-recorded | not-recorded | null | [] | [] | Half of it was proved in 2018. Whether the full statement holds decides the approximability of dozens of problems at once. | Introduced by Subhash Khot in 2002. If true, it implies that the semidefinite programming algorithms of Goemans–Williamson for MAX-CUT and of Raghavendra for every constraint satisfaction problem achieve the best approximation ratio any polynomial-time algorithm can. | - Khot, Minzer and Safra (2018) proved the 2-to-2 Games Theorem, establishing the conjecture with imperfect completeness. This is widely regarded as strong evidence for the full statement, and yields half of its consequences.
- Subexponential-time algorithms exist (Arora–Barak–Steurer), so the conjecture cannot be stre... | The conjecture is a single hypothesis from which optimal inapproximability follows for an entire class of optimisation problems. Refuting it would be equally consequential, since it would mean better approximation algorithms exist and have not been found. | published | sharp | notorious | [
"hardness-of-approximation",
"khot",
"pcp",
"sdp"
] | Subhash Khot | 2,002 | null | null | [
{
"kind": "paper",
"citation": "S. Khot, On the power of unique 2-prover 1-round games, STOC 2002, 767–775.",
"url": null,
"note": null,
"sort_order": 0
},
{
"kind": "paper",
"citation": "S. Khot, D. Minzer, M. Safra, Pseudorandom sets in Grassmann graph have near-perfect expansion, ... | {
"source_record": null,
"imported_at": null,
"status_as_of": null,
"evidence_label": null,
"note": null,
"has_full_text": null,
"original_file": null,
"sha256": null
} | null | [] | [] | {
"created_at": "2026-09-07T23:45:31.857Z",
"updated_at": "2026-09-07T23:45:31.857Z",
"published_at": "2026-09-07T23:45:35.105Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dd98c225-e193-4f41-996f-2cd41d96226b","format_issues":null,"number":27} |
1.0.0 | problem:vaught-conjecture | problem | vaught-conjecture | Vaught's conjecture | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "community-map",
"evidence": "Inferred from community field: logic-and-foundations"
}
] | Conjecture: for any complete first-order theory $T$ in a countable language, the number of countable models of $T$ up to isomorphism is either at most $\aleph_0$ or exactly $2^{\aleph_0}$. | open | {
"key": "community",
"name": "Community consensus",
"type": "problem-list",
"url": null,
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Original wording",
"notes": "Questions that are widely regarded as open but have no single canonical citation; written here from the surrounding litera... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can a countable first-order theory have strictly between ℵ₀ and 2^ℵ₀ countable models? | Posed by Robert Vaught in 1961. Under the continuum hypothesis the statement is trivially true, so the content is entirely in models of set theory where CH fails; this makes it a rare model-theoretic problem whose difficulty is not resolved by adding axioms. | - Morley (1970) proved the number of countable models is at most $\aleph_1$ or exactly $2^{\aleph_0}$, reducing the problem to excluding exactly $\aleph_1$.
- Shelah, Harrington and Makkai proved the conjecture for $\omega$-stable theories.
- The topological Vaught conjecture, for Polish group actions, remains open and... | null | published | sharp | hard | [
"countable-models",
"descriptive-set-theory",
"model-theory"
] | Robert Vaught | 1,961 | null | null | [] | {
"source_record": null,
"imported_at": null,
"status_as_of": null,
"evidence_label": null,
"note": null,
"has_full_text": null,
"original_file": null,
"sha256": null
} | null | [] | [] | {
"created_at": "2026-09-07T23:45:31.857Z",
"updated_at": "2026-09-07T23:45:31.857Z",
"published_at": "2026-09-07T23:45:35.068Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"18523fc2-e646-48e3-b41f-53c9b91925fe","format_issues":null,"number":21} |
1.0.0 | problem:wikipedia-binary-tree-rotation-distance | problem | wikipedia-binary-tree-rotation-distance | Computing binary-tree rotation distance | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | Can the minimum number of rotations transforming one rooted ordered binary tree into another be computed in polynomial time? | unassessed | {
"key": "wikipedia-computer-science",
"name": "Wikipedia — Computer science",
"type": "problem-list",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems",
"citation": "Wikipedia contributors,... | null | null | null | not-recorded | not-recorded | null | [] | [] | Can the minimum number of rotations transforming one rooted ordered binary tree into another be computed in polynomial time? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature.
Adapted and reformulated by Frontier Science from the linked Wikipedia list; attribution to Wikipedia contributors is retained under CC BY-SA 4.0. Thi... | null | null | published | sharp | hard | [
"algorithms",
"status-unassessed",
"wikipedia-selection"
] | null | null | null | null | [
{
"kind": "website",
"citation": "Wikipedia — Computer science: Computing binary-tree rotation distance (inspected 2026-09-09)",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems",
... | {
"source_record": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Wikipedia selection · status unassessed",
"note":... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:48.017Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7526269b-9b44-41df-92a3-b7ef570daef2","format_issues":null,"number":9472} |
1.0.0 | problem:wikipedia-biomolecular-homochirality | problem | wikipedia-biomolecular-homochirality | Origin of biomolecular homochirality | {
"main": "chemistry",
"sub": "chemical sciences",
"slug": "chemical-sciences",
"name": "Chemical sciences",
"original_discipline": "chemistry"
} | [] | [] | Identify how prebiotic processes generated and amplified molecular handedness into the homochirality characteristic of biological amino acids and sugars. | unassessed | {
"key": "wikipedia-chemistry",
"name": "Wikipedia — Chemistry",
"type": "problem-list",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_chemistry#Organic_chemistry_problems",
"citation": "Wikipedia contributors, List of unsolved problems in Chemistry. Adapted selection, accessed 2026-09-09... | null | null | null | not-recorded | not-recorded | null | [] | [] | Identify how prebiotic processes generated and amplified molecular handedness into the homochirality characteristic of biological amino acids and sugars. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature.
Adapted and reformulated by Frontier Science from the linked Wikipedia list; attribution to Wikipedia contributors is retained under CC BY-SA 4.0. Thi... | null | null | published | programme | hard | [
"chemical-sciences",
"status-unassessed",
"wikipedia-selection"
] | null | null | null | null | [
{
"kind": "website",
"citation": "Wikipedia — Chemistry: Origin of biomolecular homochirality (inspected 2026-09-09)",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_chemistry#Organic_chemistry_problems",
"note": null,
"sort_order": 0
},
{
"kind": "website",
"cita... | {
"source_record": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_chemistry#Organic_chemistry_problems",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Wikipedia selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current rese... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:49.571Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"adf39cb2-faf8-446a-b56b-f8f31ac4174e","format_issues":null,"number":9495} |
1.0.0 | problem:wikipedia-black-hole-information-recovery | problem | wikipedia-black-hole-information-recovery | Information recovery in black-hole evaporation | {
"main": "physics",
"sub": "mathematical physics",
"slug": "mathematical-physics",
"name": "Mathematical physics",
"original_discipline": "physics"
} | [] | [] | Explain how an evaporating black hole preserves and releases quantum information, including the relation between semiclassical Hawking radiation and a unitary microscopic description. | unassessed | {
"key": "wikipedia-physics",
"name": "Wikipedia — Physics",
"type": "problem-list",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"citation": "Wikipedia contributors, List of unsolved problems in Physics. Adapted selection, accessed 2026-09-09; CC BY-SA 4.0.",
... | null | null | null | not-recorded | not-recorded | null | [] | [] | Explain how an evaporating black hole preserves and releases quantum information, including the relation between semiclassical Hawking radiation and a unitary microscopic description. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature.
Adapted and reformulated by Frontier Science from the linked Wikipedia list; attribution to Wikipedia contributors is retained under CC BY-SA 4.0. Thi... | null | null | published | programme | hard | [
"mathematical-physics",
"status-unassessed",
"wikipedia-selection"
] | null | null | null | null | [
{
"kind": "website",
"citation": "Wikipedia — Physics: Information recovery in black-hole evaporation (inspected 2026-09-09)",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"note": null,
"sort_order": 0
},
{
"kind": "website",
"citation"... | {
"source_record": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Wikipedia selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status h... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:48.740Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cd08d7d7-8970-4b7b-b244-9ba256a66026","format_issues":null,"number":9481} |
1.0.0 | problem:wikipedia-chronology-protection | problem | wikipedia-chronology-protection | Chronology protection in quantum gravity | {
"main": "physics",
"sub": "mathematical physics",
"slug": "mathematical-physics",
"name": "Mathematical physics",
"original_discipline": "physics"
} | [] | [] | Determine whether quantum effects prevent the formation of closed timelike curves from physically admissible initial conditions, and specify the assumptions under which such protection holds. | unassessed | {
"key": "wikipedia-physics",
"name": "Wikipedia — Physics",
"type": "problem-list",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"citation": "Wikipedia contributors, List of unsolved problems in Physics. Adapted selection, accessed 2026-09-09; CC BY-SA 4.0.",
... | null | null | null | not-recorded | not-recorded | null | [] | [] | Determine whether quantum effects prevent the formation of closed timelike curves from physically admissible initial conditions, and specify the assumptions under which such protection holds. | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature.
Adapted and reformulated by Frontier Science from the linked Wikipedia list; attribution to Wikipedia contributors is retained under CC BY-SA 4.0. Thi... | null | null | published | programme | hard | [
"mathematical-physics",
"status-unassessed",
"wikipedia-selection"
] | null | null | null | null | [
{
"kind": "website",
"citation": "Wikipedia — Physics: Chronology protection in quantum gravity (inspected 2026-09-09)",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"note": null,
"sort_order": 0
},
{
"kind": "website",
"citation": "Wik... | {
"source_record": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics#Quantum_gravity",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Wikipedia selection · status unassessed",
"note": "Editorial selection inspected on 9 September 2026. Current research status h... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:48.967Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"10fae1e8-3a9a-4d08-b2f2-f4b949ac547d","format_issues":null,"number":9484} |
1.0.0 | problem:wikipedia-classical-discrete-logarithms | problem | wikipedia-classical-discrete-logarithms | Classical polynomial-time discrete logarithms | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | Given a prime $p$, a generator $g$ of $\mathbb F_p^*$ and $h\ne0$, can a classical algorithm find $x$ with $g^x=h$ in time polynomial in $\log p$? | unassessed | {
"key": "wikipedia-computer-science",
"name": "Wikipedia — Computer science",
"type": "problem-list",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems",
"citation": "Wikipedia contributors,... | null | null | null | not-recorded | not-recorded | null | [] | [] | Given a prime $p$, a generator $g$ of $\mathbb F_p^*$ and $h\ne0$, can a classical algorithm find $x$ with $g^x=h$ in time polynomial in $\log p$? | Editorial selection inspected on 9 September 2026. Current research status has not been independently reassessed; consult the original source and subsequent literature.
Adapted and reformulated by Frontier Science from the linked Wikipedia list; attribution to Wikipedia contributors is retained under CC BY-SA 4.0. Thi... | null | null | published | sharp | hard | [
"algorithms",
"status-unassessed",
"wikipedia-selection"
] | null | null | null | null | [
{
"kind": "website",
"citation": "Wikipedia — Computer science: Classical polynomial-time discrete logarithms (inspected 2026-09-09)",
"url": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems"... | {
"source_record": "https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science#Polynomial_versus_nondeterministic-polynomial_time_for_specific_algorithmic_problems",
"imported_at": "2026-09-09",
"status_as_of": "2026-09-09",
"evidence_label": "Wikipedia selection · status unassessed",
"note":... | null | [
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-09T06:35:30.942Z",
"updated_at": "2026-09-09T06:35:30.942Z",
"published_at": "2026-09-09T06:35:48.078Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c19b8d3b-8887-40e9-946a-dfe4f10c6a60","format_issues":null,"number":9473} |
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