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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1,292 | NT-084 | Bunyakovsky Conjecture | Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes? | Proposed by Viktor Bunyakovsky in 1857, this generalizes Dirichlet's theorem on primes in arithmetic progressions. It states that if polynomial $f(x)$ has integer coefficients, positive leading coefficient, is irreducible over integers, and has no common prime divisor of all its values $f(n)$ for positive integers $n$,... | 5 | open | null | null | 1 | null | 512 | 41 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,293 | NT-085 | Dickson's Conjecture | Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions? | Proposed by Leonard Eugene Dickson in 1904, this generalizes Dirichlet's theorem and implies many prime conjectures. For linear forms $a_1 + b_1 n, \ldots, a_k + b_k n$ with each $b_i \geq 1$, if no congruence condition forces a composite, then infinitely many $n$ exist making all forms simultaneously prime. This would... | 5 | partially_solved | null | null | 1 | null | 445 | 34 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,294 | NT-086 | Brocard's Conjecture (Prime Gaps) | Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$? | Proposed by Henri Brocard in 1904, this conjecture concerns the density of primes near perfect squares. For consecutive primes $p_n$ and $p_{n+1}$, Brocard conjectured there are always at least 4 primes in the interval $(p_n^2, p_{n+1}^2)$, except for the cases $(2^2, 3^2)$ which contains only one prime (5). Verified c... | 4 | partially_solved | null | null | 1 | null | 398 | 29 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | The interval between 2 squared and 3 squared only refutes the missing n>=2 qualification. The standard Brocard conjecture remains open.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The cited partial result or literal observation is retained. | [
{
"url": "https://mathworld.wolfram.com/BrocardsConjecture.html",
"label": "Eric W. Weisstein, Brocard's Conjecture, MathWorld."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
... | 2026-09-27T00:00:00 | null | {
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"checked_at": "2026-09-27T00:00:00",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,295 | NT-087 | Agoh-Giuga Conjecture | Is $p$ prime if and only if $pB_{p-1} \equiv -1 \pmod{p}$ for the Bernoulli number $B_{p-1}$? | This conjecture combines work of Takashi Agoh (1990) and Giuseppe Giuga (1950), providing a primality criterion via Bernoulli numbers. Bernoulli numbers $B_n$ appear in number theory and analysis. The conjecture states: $p$ is prime iff $pB_{p-1} \equiv -1 \pmod{p}$. The forward direction is known (if $p$ prime, the co... | 4 | partially_solved | null | null | 1 | null | 334 | 25 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,296 | NT-088 | Elliott-Halberstam Conjecture | Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)? | Proposed in 1968, this strengthens the Bombieri-Vinogradov theorem about primes in arithmetic progressions. For most moduli $q < x^\theta$, the primes are equidistributed among valid residue classes. Bombieri-Vinogradov proves this for $\theta < 1/2$. Elliott-Halberstam conjectures it holds for any $\theta < 1$. This w... | 5 | partially_solved | null | null | 1 | null | 412 | 32 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,297 | ALG-001 | Birch–Tate Conjecture | Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function? | The Birch–Tate conjecture connects algebraic K-theory to number theory. For a number field $F$, it relates the order of the center of the Steinberg group $\text{St}(\mathcal{O}_F)$ (where $\mathcal{O}_F$ is the ring of integers) to special values of the Dedekind zeta function $\zeta_F(s)$ at $s = -1$. The conjecture pr... | 5 | partially_solved | null | null | 4 | null | 245 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,300 | ALG-004 | Crouzeix's Conjecture | Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range? | Michel Crouzeix conjectured in 2004 that for any $n \times n$ complex matrix $A$ and any function $f$ analytic on the numerical range $W(A) = \{\langle Ax, x \rangle : \|x\| = 1\}$, the matrix norm satisfies $\|f(A)\| \leq 2\|f\|_{W(A)}$. The constant 2 is conjectured to be optimal. Crouzeix proved the bound with const... | 4 | solved | null | null | 4 | null | 278 | 21 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | Lorist and Schwenninger (August 2026) claim the constant-2 Crouzeix bound; Shanmu Jin independently announced a proof in July 2026. These are unrefereed preprints. The analytic formulation uses functions holomorphic on a neighborhood of the numerical range.
**What remains.** Independent expert verification and peer re... | [
{
"url": "https://arxiv.org/abs/2608.03841",
"label": "Emiel Lorist and Felix Schwenninger, A solution to Crouzeix’s conjecture (2026)"
},
{
"url": "https://www.preprints.org/manuscript/202607.1919",
"label": "Shanmu Jin, The Numerical Range Is a 2-Spectral Set (2026)"
},
{
"url": "https... | 2026-09-27T00:00:00 | Emiel Lorist and Felix Schwenninger, A solution to Crouzeix’s conjecture (2026): https://arxiv.org/abs/2608.03841
Shanmu Jin, The Numerical Range Is a 2-Spectral Set (2026): https://www.preprints.org/manuscript/202607.1919 | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "unrefereed_preprint"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,302 | ALG-006 | Perfect Cuboid | Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal? | A perfect cuboid would have integer values for all of: three edge lengths $a, b, c$, three face diagonals $\sqrt{a^2+b^2}, \sqrt{b^2+c^2}, \sqrt{c^2+a^2}$, and the space diagonal $\sqrt{a^2+b^2+c^2}$. This is the 3D generalization of the Pythagorean triple problem (which has infinitely many solutions). Despite extensiv... | 3 | partially_solved | null | null | 4 | null | 423 | 35 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,305 | ALG-009 | Zauner's Conjecture (SIC-POVM) | Do symmetric informationally complete POVMs exist in all dimensions? | Zauner's conjecture, central to quantum information theory, asks whether SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite-dimensional Hilbert spaces. A SIC-POVM in dimension $d$ consists of $d^2$ pure quantum states with pairwise fidelity $1/(d+1)$, forming a regular ... | 4 | partially_solved | null | null | 4 | null | 298 | 26 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,308 | ALG-012 | Andrews–Curtis Conjecture | Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves? | The Andrews–Curtis conjecture, proposed in 1965, concerns group presentations. A balanced presentation has the same number of generators and relators. The trivial presentation is $\langle x \mid x \rangle$. Nielsen transformations on relators include: replacing relator $r$ with $r^{-1}$, with $rs$ for another relator $... | 4 | partially_solved | null | null | 4 | null | 289 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,310 | ALG-014 | Herzog–Schönheim Conjecture | Can a finite system of left cosets forming a partition of a group have distinct indices? | The Herzog–Schönheim conjecture states: if left cosets $g_iH_i$ of subgroups $H_i$ partition a group $G$, then at least two indices $[G:H_i]$ must be equal. Equivalently, you cannot partition a group using cosets of subgroups with all different indices. The conjecture is verified for many cases: finite abelian groups, ... | 4 | partially_solved | null | null | 4 | null | 198 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,318 | ANA-006 | Navier-Stokes Regularity | Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time? | One of the seven Millennium Prize Problems ($1M prize). The 3D Navier-Stokes equations govern fluid flow: $\partial_t u + (u \cdot \nabla)u = \nu \Delta u - \nabla p + f$ with $\nabla \cdot u = 0$. Given smooth initial conditions and forcing, do solutions remain smooth globally, or can finite-time singularities develop... | 5 | partially_solved | null | null | 9 | null | 892 | 67 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,319 | COMB-001 | 1/3–2/3 Conjecture | Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions? | For a finite partially ordered set (poset) that is not totally ordered, the 1/3–2/3 conjecture asks: do there always exist elements $x$ and $y$ such that the probability $x$ appears before $y$ in a uniformly random linear extension is strictly between 1/3 and 2/3? Linear extensions are total orderings consistent with t... | 4 | partially_solved | null | null | 2 | null | 234 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,320 | COMB-002 | Lonely Runner Conjecture | If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time? | Proposed by J. M. Wills in 1967, this conjecture concerns runners on a unit-length circular track with distinct constant speeds. A runner is "lonely" if all other runners are at distance at least $1/k$ away. The conjecture states every runner is lonely at some time. Verified for $k \leq 7$ runners. The problem has refo... | 4 | partially_solved | null | null | 2 | null | 312 | 26 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,321 | COMB-003 | Union-Closed Sets Conjecture | For a finite family of sets closed under unions, must some element appear in at least half the sets? | Frankl's union-closed sets conjecture (1979) states: if a finite family $\mathcal{F}$ of sets is closed under pairwise unions (i.e., $A, B \in \mathcal{F} \Rightarrow A \cup B \in \mathcal{F}$), then there exists an element appearing in at least $|\mathcal{F}|/2$ sets. The conjecture is verified for many special cases:... | 4 | partially_solved | null | null | 2 | null | 387 | 31 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,322 | COMB-004 | No-Three-in-Line Problem | What is the maximum number of points in an $n \times n$ grid with no three collinear? | The no-three-in-line problem asks for $g(n)$, the maximum number of points that can be placed in an $n \times n$ grid such that no three are collinear. Dudeney (1917) conjectured $g(n) = 2n$ for all $n$. Known values: $g(3) = 4, g(4) = 8, g(5) = 10, g(6) = 12$, and computational results extend further. For large $n$, E... | 3 | partially_solved | null | null | 2 | null | 298 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,324 | COMB-006 | Sunflower Conjecture | For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$? | Erdős and Rado (1960) defined an $r$-sunflower as a collection of $r$ sets $A_1, \ldots, A_r$ with common intersection $C$ (the core) such that the sets $A_i \setminus C$ are pairwise disjoint (the petals). Their theorem: any family of size-$k$ sets with at least $k! \cdot r^k$ members contains an $r$-sunflower. The su... | 4 | partially_solved | null | null | 2 | null | 367 | 29 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,327 | GRAPH-003 | Cycle Double Cover Conjecture | Does every bridgeless graph have a collection of cycles covering each edge exactly twice? | The cycle double cover conjecture states: every bridgeless graph (no bridge edges) has a cycle double cover—a collection of cycles such that each edge appears in exactly two cycles. Proposed by Szekeres (1973) and Seymour (1979), this is equivalent to several other conjectures in graph theory. Known for planar graphs (... | 4 | solved | null | null | 3 | null | 312 | 25 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | A proof of the cycle double cover conjecture was announced in July 2026. Sang-il Oum’s exposition presents the proof for every bridgeless graph. This is a reported resolution in an unrefereed preprint, not a claim of completed peer review.
**What remains.** Independent expert verification and publication of the announ... | [
{
"url": "https://arxiv.org/abs/2607.16356",
"label": "Sang-il Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition (2026)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates ... | 2026-09-27T00:00:00 | Sang-il Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition (2026): https://arxiv.org/abs/2607.16356 | {
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"checked_at": "2026-09-27T00:00:00",
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} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,328 | GRAPH-004 | Erdős–Hajnal Conjecture | For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets? | The Erdős–Hajnal conjecture (1977) asks: for any graph $H$, is there $\delta > 0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or independent set of size at least $n^\delta$? This would dramatically strengthen Ramsey theory for hereditary graph classes. For general graphs, Ramsey theor... | 5 | partially_solved | null | null | 3 | null | 289 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,329 | GRAPH-005 | Lovász Conjecture | Does every finite connected vertex-transitive graph have a Hamiltonian path? | Proposed by László Lovász in 1969, this conjecture states that every finite connected vertex-transitive graph (graph with transitive automorphism group) contains a Hamiltonian path. A stronger version asks for a Hamiltonian cycle. The conjecture is verified for Cayley graphs (Rapaport-Strasser, 1985 for primes; Marušič... | 4 | open | null | null | 3 | null | 267 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,330 | GRAPH-006 | Hadwiger–Nelson Problem | What is the chromatic number of the plane with unit distance graph coloring? | The Hadwiger–Nelson problem asks: what is the minimum number of colors needed to color the plane such that no two points at distance exactly 1 have the same color? This is equivalent to finding the chromatic number of the unit distance graph in $\mathbb{R}^2$. It has been known since 1950 that $4 \leq \chi \leq 7$. In ... | 4 | partially_solved | null | null | 3 | null | 421 | 35 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,331 | TOP-001 | Unknotting Problem | Can unknots be recognized in polynomial time? | The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a given knot diagram represents the unknot (a circle with no actual knots). A knot diagram is a 2D projection of a 3D knot with crossing information. The problem is known to be in NP (a certificate is a sequence of Reidemeister... | 4 | partially_solved | null | null | 7 | null | 334 | 27 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,332 | TOP-002 | Borel Conjecture | Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups? | The Borel conjecture states: if two aspherical closed manifolds (manifolds with contractible universal cover) have isomorphic fundamental groups, then they are homeomorphic. An aspherical manifold has all higher homotopy groups trivial, so its topology is determined by $\pi_1$. The conjecture is a topological rigidity ... | 5 | partially_solved | null | null | 7 | null | 278 | 22 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,333 | TOP-003 | Volume Conjecture | Do quantum invariants of knots relate asymptotically to hyperbolic volume? | The volume conjecture, proposed by Kashaev (1997) and generalized by Murakami-Murakami, relates quantum topology to hyperbolic geometry. For a hyperbolic knot $K$ in $S^3$, let $J_N(K; q)$ be the colored Jones polynomial at $q = e^{2\pi i/N}$. The conjecture states: $\lim_{N \to \infty} \frac{2\pi \log|J_N(K; e^{2\pi i... | 5 | partially_solved | null | null | 7 | null | 245 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,334 | TOP-004 | Novikov Conjecture | Are certain combinations of Pontryagin classes homotopy invariant? | The Novikov conjecture, proposed by Sergei Novikov in 1965, is a fundamental problem in topology and differential geometry. For a closed oriented manifold $M$ with fundamental group $\pi$, certain rational linear combinations of Pontryagin classes evaluated on the fundamental class should be homotopy invariants when pu... | 5 | partially_solved | null | null | 7 | null | 312 | 25 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,335 | GEOM-007 | Kakeya Conjecture | Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$? | A Kakeya set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. The Kakeya conjecture states such sets must have full Hausdorff and Minkowski dimension $n$. In $\mathbb{R}^2$, Kakeya sets can have measure zero (Davies 1971) but must have Hausdorff dimension 2 (proven). For $n \geq 3$,... | 5 | partially_solved | null | null | 6 | null | 289 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,336 | GEOM-008 | Illumination Problem | Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources? | The illumination problem (or Hadwiger's problem) asks: what is the minimum number of light sources (point sources or directions) needed to illuminate the entire boundary of any convex body in $\mathbb{R}^n$? A point on the boundary is illuminated if the ray from the light source to that point does not intersect the int... | 4 | open | null | null | 6 | null | 234 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,338 | DYN-002 | MLC Conjecture | Is the Mandelbrot set locally connected? | The MLC (Mandelbrot set is Locally Connected) conjecture asks whether the famous Mandelbrot set—the set of complex parameters $c$ for which the iteration $z_{n+1} = z_n^2 + c$ (starting from $z_0 = 0$) remains bounded—is locally connected. Local connectivity would mean every point has arbitrarily small connected neighb... | 5 | partially_solved | null | null | 9 | null | 398 | 32 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,339 | DYN-003 | Weinstein Conjecture | Does every regular compact contact-type level set carry a periodic orbit? | The Weinstein conjecture, proposed by Alan Weinstein in 1978, states: every regular compact contact-type level set of a Hamiltonian on a symplectic manifold carries at least one periodic orbit of the Hamiltonian flow. In more geometric terms, on a compact contact manifold, the Reeb vector field has at least one closed ... | 5 | partially_solved | null | null | 9 | null | 256 | 20 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,340 | DYN-004 | Birkhoff Conjecture | If a billiard table is strictly convex and integrable, must its boundary be an ellipse? | The Birkhoff conjecture concerns dynamical billiards: if a strictly convex billiard table in the plane is integrable (has a complete set of integrals of motion), then its boundary must be an ellipse. Elliptical billiards are known to be integrable (Birkhoff, 1927). The conjecture asks if they are the only such tables. ... | 5 | partially_solved | null | null | 9 | null | 289 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,341 | ALGGEOM-001 | Abundance Conjecture | If the canonical bundle of a variety is nef, must it be semiample? | The abundance conjecture is a central problem in birational geometry and minimal model theory. For a projective variety $X$ with Kawamata log terminal singularities, if the canonical bundle $K_X$ is nef (numerically effective—has non-negative intersection with all curves), the conjecture states $K_X$ must be semiample ... | 5 | open | null | null | 5 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,343 | LOGIC-001 | Vaught Conjecture | Is the number of countable models of a complete first-order theory finite, $\aleph_0$, or $2^{\aleph_0}$? | The Vaught conjecture, proposed by Robert Vaught in 1961, is a fundamental problem in model theory. For a complete first-order theory in a countable language, the number of countable models (up to isomorphism) must be either finite, countably infinite ($\aleph_0$), or continuum ($2^{\aleph_0}$). In other words, there c... | 5 | partially_solved | null | null | 4 | null | 298 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,344 | LOGIC-002 | Cherlin-Zilber Conjecture | Is every simple group with $\aleph_0$-stable theory an algebraic group over an algebraically closed field? | The Cherlin-Zilber conjecture concerns the classification of simple groups in model theory. It states: every infinite simple group whose first-order theory is stable in $\aleph_0$ (countably stable) is isomorphic to a simple algebraic group over an algebraically closed field. The conjecture connects abstract model-theo... | 5 | partially_solved | null | null | 4 | null | 245 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,346 | GEOM-009 | Yang-Mills Existence and Mass Gap | Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D? | One of the seven Millennium Prize Problems ($1M prize). The Yang-Mills equations describe the behavior of elementary particles using non-Abelian gauge theory, fundamental to the Standard Model of particle physics. The problem asks two questions: (1) Does a mathematically rigorous quantum Yang-Mills theory exist in 4-di... | 5 | open | null | null | 6 | null | 567 | 47 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,347 | ST-001 | Partition Principle Implies Axiom of Choice | Does the partition principle (PP) imply the axiom of choice (AC)? | The partition principle states that for every partition of a set, there exists a set that contains exactly one element from each cell of the partition. The axiom of choice states that for every collection of nonempty sets, there exists a choice function selecting one element from each set. While AC clearly implies PP, ... | 4 | open | null | null | 10 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,348 | ST-002 | Woodin's GCH below Strongly Compact Cardinals | Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere? | Posed by W. Hugh Woodin, this problem asks whether local instances of the generalized continuum hypothesis (GCH) can force global instances. A strongly compact cardinal is a large cardinal with strong reflection properties. The question explores whether GCH holding below such a cardinal must propagate throughout the un... | 5 | open | null | null | 10 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,349 | ST-003 | GCH and Diamond Principle | Does the generalized continuum hypothesis entail the diamond principle $\diamondsuit(E_{\text{cf}(\lambda)}^{\lambda^+})$ for every singular cardinal $\lambda$? | The diamond principle is a combinatorial principle asserting the existence of certain prediction sequences. For singular cardinals (cardinals not equal to their own cofinality), the relationship between GCH and diamond principles is subtle. While diamond holds at successor cardinals under GCH, its behavior at successor... | 5 | open | null | null | 10 | null | 156 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,350 | ST-004 | GCH and Suslin Trees | Does the generalized continuum hypothesis imply the existence of an $\aleph_2$-Suslin tree? | A Suslin tree is a tree of height $\omega_1$ with no uncountable chains or antichains. An $\aleph_2$-Suslin tree is the analogous structure at the next cardinal level. While Suslin trees at $\aleph_1$ can exist under certain axioms, their existence at $\aleph_2$ under GCH is unknown. This problem connects cardinal arit... | 4 | open | null | null | 10 | null | 167 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,352 | ST-006 | Ultimate Core Model | Does there exist an ultimate core model containing all large cardinals? | Core models are canonical inner models of set theory that approximate the entire universe while being more tractable. The search for an ultimate core model—one encompassing all large cardinal properties—is a central goal of modern set theory. Such a model would unify our understanding of large cardinals and provide a f... | 5 | open | null | null | 10 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,353 | ST-007 | Woodin's Ω-Conjecture | If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem? | Proposed by W. Hugh Woodin, this conjecture connects large cardinals with logic. Ω-logic is a strong logic using Woodin cardinals to define semantic validity. The conjecture asserts that under the assumption of a proper class of Woodin cardinals, Ω-logic becomes complete in a generalized sense—every Ω-valid sentence ha... | 5 | open | null | null | 10 | null | 145 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,354 | ST-008 | Strongly Compact vs Supercompact Cardinals | Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal? | Strongly compact cardinals and supercompact cardinals are both large cardinal notions with powerful reflection properties. Supercompact cardinals are known to be stronger, but whether their consistency strength is strictly greater than strongly compact cardinals remains open. This problem probes the fine structure of t... | 5 | open | null | null | 10 | null | 167 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,355 | ST-009 | Jónsson Algebra on ℵ_ω | Does there exist a Jónsson algebra on $\aleph_\omega$? | A Jónsson algebra is an algebraic structure with no proper subalgebra of the same cardinality. The existence of Jónsson algebras on various cardinals connects algebra with set theory. For $\aleph_\omega$ (the $\omega$-th infinite cardinal), existence remains unknown. A positive answer would provide new insights into th... | 4 | open | null | null | 10 | null | 134 | 10 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,356 | ST-010 | Open Coloring Axiom and Continuum Hypothesis | Is the open coloring axiom (OCA) consistent with $2^{\aleph_0} > \aleph_2$? | The open coloring axiom is a combinatorial principle with powerful consequences for the structure of the real line. It is known to be consistent with $2^{\aleph_0} = \aleph_2$, but consistency with larger values of the continuum is unknown. This problem explores the interaction between partition properties and cardinal... | 4 | open | null | null | 10 | null | 156 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,357 | ST-011 | Reinhardt Cardinals without Choice | Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist? | A Reinhardt cardinal would witness an elementary embedding from the universe of all sets (V) to itself. Kunen proved such embeddings cannot exist with the axiom of choice. However, without AC, the question remains open. Reinhardt cardinals would be the strongest large cardinal notion, transcending the usual hierarchy. ... | 5 | open | null | null | 10 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,358 | GAME-001 | Sudoku: Unique Solution Puzzles | How many Sudoku puzzles have exactly one solution? | Standard 9×9 Sudoku grids can be filled in approximately 6.67 × 10²¹ ways. A puzzle is a partial filling with a unique completion. Despite extensive computer searches, the exact count of puzzles with unique solutions remains unknown. This combinatorial problem involves constraints, symmetry breaking, and counting techn... | 2 | partially_solved | null | null | 2 | null | 892 | 67 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,359 | GAME-002 | Sudoku: Minimal Puzzles Count | How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)? | A minimal Sudoku puzzle cannot have any clue removed without losing uniqueness. While we know examples with as few as 17 clues, the total count of minimal puzzles is unknown. This problem combines enumeration with the structure of constraint systems. The answer would deepen our understanding of puzzle difficulty, minim... | 2 | partially_solved | null | null | 2 | null | 678 | 51 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,360 | GAME-003 | Maximum Givens in Minimal Sudoku | What is the maximum number of givens for a minimal Sudoku puzzle? | While minimal puzzles can have as few as 17 givens, the upper bound is unknown. A puzzle with many givens can still be minimal if each clue is essential. Computer searches have found minimal puzzles with around 40 givens, but no theoretical maximum is known. This question explores the relationship between redundancy, m... | 2 | partially_solved | null | null | 2 | null | 567 | 43 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,361 | GAME-004 | Tic-Tac-Toe Winning Dimension | Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy? | Classic tic-tac-toe is a draw with perfect play. In higher dimensions (n^d game), questions become more complex. The Hales-Jewett theorem guarantees that for any fixed line length n, there exists a dimension d where the first player wins. But finding the exact threshold dimension for each n remains open. This connects ... | 3 | partially_solved | null | null | 2 | null | 445 | 34 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,362 | GAME-005 | Perfect Chess | What is the outcome of a perfectly played game of chess? | Chess is a finite deterministic game, so theoretically one of three outcomes holds with perfect play: White wins, Black wins, or draw. Despite centuries of play and powerful computers, we don't know which. The game tree has approximately 10⁴⁷ positions, far beyond exhaustive analysis. Current evidence suggests a draw, ... | 3 | open | null | null | 2 | null | 1,534 | 112 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,363 | GAME-006 | Perfect Komi in Go | What is the perfect value of komi (compensation points) in Go? | In Go, komi compensates the second player (White) for Black's first-move advantage. Professional play uses 6.5 or 7.5 points. But what value makes the game perfectly fair with optimal play? Go's complexity (10¹⁷⁰ legal positions) prevents exhaustive analysis. AI like AlphaGo suggest small adjustments, but perfect komi ... | 3 | open | null | null | 2 | null | 789 | 58 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,364 | GAME-007 | Cap Set Problem | What is the largest possible cap set in $n$-dimensional affine space over the three-element field? | A cap set is a collection of points with no three in a line (in the game SET, cards with no valid set). In the affine space $\mathbb{F}_3^n$, the maximum cap set size is conjectured to be $c^n$ for some constant c < 3. The best bounds are $2.756^n$ (Ellenberg-Gijswijt, 2016). Determining the precise growth rate connect... | 4 | partially_solved | null | null | 2 | null | 356 | 28 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,365 | GAME-008 | Octal Games Periodicity | Are the nim-sequences of all finite octal games eventually periodic? | Octal games are impartial combinatorial games defined by simple rules encoded in octal notation. Their nim-values (Grundy numbers) determine optimal play. For some octal games, the nim-sequence is eventually periodic; for others, patterns are elusive. Whether all finite octal games have eventually periodic nim-sequence... | 3 | open | null | null | 2 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,366 | GAME-009 | Grundy's Game Periodicity | Is the nim-sequence of Grundy's game eventually periodic? | Grundy's game: split a heap of n beans into two unequal heaps; last player to move wins. The nim-value sequence starts 0,1,0,2,1,3,2,1,0,4,... but no period has been found despite extensive computation. Whether it's eventually periodic (or even computable) is open. This specific game has resisted analysis for decades, ... | 3 | open | null | null | 2 | null | 278 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,367 | GAME-010 | Rendezvous Problem | What is the optimal strategy for two agents to meet on a network without communication? | The rendezvous problem asks: how should two agents move on a graph to minimize expected meeting time, when they can't communicate and may not know the graph structure? Variants include symmetric/asymmetric information, labeled/unlabeled nodes, and different graph families. Optimal strategies are known for some simple c... | 3 | partially_solved | null | null | 2 | null | 312 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,369 | GEOM-010 | Kissing Number Problem | What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, 8, and 24? | The kissing number is known exactly only in dimensions 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196,560). The problem asks for exact values in other dimensions. In dimension 3, twelve spheres can kiss a central sphere (with centers forming an icosahedron). Dimensions 8 and 24 have exceptional symmetries related t... | 4 | partially_solved | null | null | 6 | null | 534 | 41 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,372 | GEOM-013 | Tammes Problem | For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere? | The Tammes problem asks: how should n points be arranged on a sphere to maximize the minimum distance between any pair? This is equivalent to packing n spherical caps on a sphere. Solutions are known for n ≤ 14 and n = 24 (related to exceptional geometries). For other n, only bounds and computational results exist. The... | 3 | partially_solved | null | null | 6 | null | 245 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,373 | GEOM-014 | Carathéodory Conjecture | Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points? | An umbilical point on a surface is where the two principal curvatures are equal (the surface curves equally in all directions, like on a sphere). Carathéodory conjectured that every smooth convex closed surface must have at least two umbilic points. A sphere has infinitely many (every point), but most surfaces should h... | 4 | open | null | null | 6 | null | 312 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,374 | GEOM-015 | Cartan-Hadamard Conjecture | Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)? | The classical isoperimetric inequality states that among all regions with fixed perimeter in Euclidean space, the circle (or sphere) encloses maximum area (or volume). The Cartan-Hadamard conjecture asks whether this inequality holds in spaces of nonpositive curvature. Proven in dimensions 2, 3, and 4, but open in high... | 4 | partially_solved | null | null | 6 | null | 267 | 20 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,375 | GEOM-016 | Chern's Conjecture (Affine Geometry) | Does the Euler characteristic of a compact affine manifold vanish? | An affine manifold is a manifold with an atlas whose transition functions are affine transformations. Chern conjectured that any closed (compact, boundaryless) affine manifold must have Euler characteristic zero. The conjecture is true for many special cases but remains open in general. This would be a fundamental cons... | 4 | open | null | null | 6 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,376 | GEOM-017 | Hopf Conjectures | What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds? | The Hopf conjectures are a collection of problems relating the curvature of a manifold to its Euler characteristic. One version: does a positively curved even-dimensional manifold have positive Euler characteristic? Another: does a negatively curved manifold have zero Euler characteristic? These would generalize the Ga... | 5 | partially_solved | null | null | 6 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,377 | GEOM-018 | Yau's Conjecture on First Eigenvalue | Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$? | This conjecture by Shing-Tung Yau concerns minimal surfaces (soap-film-like surfaces) embedded in spheres. The Laplace-Beltrami operator generalizes the Laplacian to curved spaces. Yau conjectured that the first eigenvalue equals the dimension n for minimal hypersurfaces in the (n+1)-sphere. This would provide a sharp ... | 5 | partially_solved | null | null | 6 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,378 | GEOM-019 | Hadwiger Conjecture (Covering) | Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies? | Hadwiger conjectured that any convex body in n dimensions can be covered by at most 2ⁿ smaller copies that are scaled-down versions (homotheties with positive ratio). Proven only for n ≤ 3. For n=2, four copies suffice (proven by Levi). For n=3, eight copies suffice (Hadwiger's original proof). Higher dimensions remain... | 4 | partially_solved | null | null | 6 | null | 298 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,379 | GEOM-020 | Happy Ending Problem | What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon? | The Happy Ending problem (named for the romance between Erdős and Szekeres who solved special cases) asks: how many points in general position (no three collinear) force the existence of n points forming a convex n-gon? Known: g(3)=3, g(4)=5, g(5)=9. Erdős-Szekeres proved $2^{n-2} + 1 \leq g(n) \leq \binom{2n-4}{n-2} +... | 4 | partially_solved | null | null | 6 | null | 345 | 27 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,380 | GEOM-021 | Heilbronn Triangle Problem | What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine? | Heilbronn asked: place n points in a unit square to maximize the minimum triangle area. Trivially, the minimum area is ≤ 2/n. Heilbronn conjectured it's O(1/n²). Komlos-Pintz-Szemeredi showed it's actually Θ((log n)/n²), disproving the conjecture. However, the exact constant is unknown, and tight bounds remain elusive.... | 4 | open | null | null | 6 | null | 223 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
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"slug": "geometry",
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,381 | GEOM-022 | Kalai's 3^d Conjecture | Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces? | Gil Kalai conjectured that centrally symmetric polytopes (symmetric under reflection through the origin) must have many faces—at least 3^d for dimension d. The d-cube achieves this bound exactly. Proved for d ≤ 4. Higher dimensions remain open. This would be a fundamental constraint on the combinatorial complexity of s... | 4 | partially_solved | null | null | 6 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
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"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,382 | GEOM-023 | Orchard-Planting Problem | What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane? | An orchard-planting problem asks: arrange n points (trees) to maximize the number of lines containing exactly 3 points (rows). For n points, at most n(n-1)/6 such lines are possible (by counting). Some configurations achieve this bound or come close. The problem asks for the exact maximum for each n. Solutions are know... | 3 | partially_solved | null | null | 6 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,383 | GEOM-024 | Unit Distance Problem | How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane? | Erdős asked: what's the maximum number of unit-distance pairs among n points in the plane? Trivially at most n(n-1)/2. Known: the maximum is Θ(n^(4/3)) (lower bound by Erdős, upper by Spencer-Szemerédi-Trotter). But the exact exponent is unknown—it could be n^(4/3), n^(3/2), or something between. Determining this conne... | 4 | partially_solved | null | null | 6 | null | 267 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,384 | GEOM-025 | Bellman's Lost-in-a-Forest Problem | What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation? | You're lost in a forest (a region with known shape but unknown location and orientation). What path guarantees you'll reach the edge? For a circle of radius 1, a path of length ≤ 2 + π/3 ≈ 3.05 suffices. For a square, the answer is unknown. For general convex regions, the problem is wide open. This classic problem in g... | 3 | partially_solved | null | null | 6 | null | 423 | 33 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
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"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,385 | GEOM-026 | Borromean Rings Question | Can three unknotted space curves (not all circles) be arranged as Borromean rings? | Borromean rings are three linked loops where removing any one unlinks the other two. Classical Borromean rings use circles, but perfect circular realization is impossible (proved). Can non-circular unknotted curves realize this linking pattern? This question connects knot theory, topology, and geometry. While Borromean... | 3 | partially_solved | null | null | 6 | null | 312 | 24 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | A noncircular Borromean realization settles the literal existence question, but does not settle stronger questions about prescribed curve shapes. The intended problem needs clarification.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The cited partial ... | [
{
"url": "https://en.wikipedia.org/wiki/Borromean_rings",
"label": "Borromean rings overview."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
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] | 2026-09-27T00:00:00 | null | {
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1,386 | GEOM-027 | Danzer's Problem | Do Danzer sets of bounded density or bounded separation exist? | A Danzer set is a set of points in the plane such that every convex region of area 1 contains at least one point. Danzer asked: can such a set have bounded density (points per unit area) or bounded separation (minimum distance between points)? Both properties would mean the points are "well-distributed." While Danzer s... | 4 | open | null | null | 6 | null | 201 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
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"order_index": 6,
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,388 | GRAPH-001 | Brouwer's Conjecture on Graph Laplacians | Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges? | Brouwer conjectured an upper bound for the sum of the k largest eigenvalues of the Laplacian matrix of a graph in terms of the number of edges. The Laplacian matrix encodes graph structure and has deep connections to spectral graph theory. This conjecture would provide fundamental insights into the relationship between... | 4 | partially_solved | null | null | 3 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,389 | GRAPH-002 | Eternal Domination vs Domination Number | Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number? | The dominating number γ(G) is the minimum size of a dominating set. The eternal dominating number γ∞(G) arises from a game where guards on vertices must respond to attacks. The question asks if these can equal each other while being smaller than the clique covering number (minimum number of cliques needed to cover all ... | 3 | open | null | null | 3 | null | 156 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,390 | GRAPH-003 | Graham's Pebbling Conjecture | Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers? | Graph pebbling is a combinatorial game where pebbles are moved on vertices according to specific rules. Graham conjectured that the pebbling number (minimum pebbles needed to guarantee placing one on any target vertex) of a Cartesian product G × H is at least π(G) × π(H). Despite progress on special cases like products... | 4 | open | null | null | 3 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,391 | GRAPH-004 | Meyniel's Conjecture on Cop Number | Is the cop number of a connected n-vertex graph $O(\sqrt{n})$? | The cop number is the minimum number of cops needed to guarantee catching a robber in a pursuit game on a graph. Meyniel conjectured that for any connected graph with n vertices, the cop number is at most O(√n). The best known upper bound is O(n/log n). This problem connects graph theory with algorithmic game theory an... | 4 | partially_solved | null | null | 3 | null | 267 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,392 | GRAPH-005 | Graph Coloring Game Monotonicity | If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors? | In the graph coloring game, two players alternately color vertices with k colors, trying to create (Alice) or avoid (Bob) a proper coloring. Intuitively, having more colors should make Alice's task easier. However, whether winning with k colors implies winning with k+1 colors is surprisingly still open. This problem pr... | 3 | open | null | null | 3 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,393 | GRAPH-006 | 1-Factorization Conjecture | Does every k-regular graph on 2n vertices admit a 1-factorization when k ≥ n (or k ≥ n-1 for even n)? | A 1-factor is a perfect matching, and a 1-factorization is a partition of edges into 1-factors. The conjecture states that sufficiently regular graphs can be decomposed into perfect matchings. This would generalize classical results on complete graphs. Proven for many special cases, but the general statement remains op... | 4 | partially_solved | null | null | 3 | null | 201 | 16 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | The cited 1-factorization theorem applies to sufficiently large graphs. The imported record omits that qualification, so the cited result alone does not justify a solved label for every finite order.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The ci... | [
{
"url": "https://arxiv.org/abs/1401.4159",
"label": "B. Csaba et al., Proof of the 1-factorization and Hamilton decomposition conjectures, Mem. AMS 244 (2016); arXiv:1401.4159."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4... | 2026-09-27T00:00:00 | null | {
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1,394 | GRAPH-007 | Perfect 1-Factorization Conjecture | Does every complete graph on an even number of vertices admit a perfect 1-factorization? | A perfect 1-factorization of a complete graph K₂ₙ is a 1-factorization where the union of any two 1-factors forms a Hamiltonian cycle. Such structures have beautiful symmetry and applications to combinatorial designs. While perfect 1-factorizations are known for many n (especially powers of 2 and small cases), a genera... | 4 | partially_solved | null | null | 3 | null | 234 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,395 | GRAPH-008 | Cereceda's Conjecture | For k-degenerate graphs, can any (k+2)-coloring be transformed to any other in polynomial steps via single-vertex recolorings? | Cereceda's conjecture concerns the diameter of the reconfiguration graph of graph colorings. It asks whether the shortest sequence of single-vertex recolorings transforming one coloring to another is polynomially bounded for degenerate graphs. This connects graph coloring with reconfiguration problems—a growing area st... | 4 | partially_solved | null | null | 3 | null | 167 | 13 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | Known polynomial bounds for fixed degeneracy settle the weak wording here, while the standard Cereceda conjecture asks for a uniform quadratic bound. Retain partial progress for the intended conjecture.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The... | [
{
"url": "https://arxiv.org/abs/1903.05619",
"label": "N. Bousquet and M. Heinrich, A polynomial version of Cereceda's conjecture, J. Combin. Theory Ser. B 155 (2022), doi:10.1016/j.jctb.2022.01.006; arXiv:1903.05619."
},
{
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1,396 | GRAPH-009 | Earth-Moon Problem | What is the maximum chromatic number of biplanar graphs? | A graph is biplanar if it can be drawn on two parallel planes (Earth and Moon) with edges possibly crossing between planes but not within each plane. The Earth-Moon problem asks for the maximum chromatic number of such graphs. Known bounds are 12 ≤ χ ≤ 16. This problem combines planarity concepts with multilayer graph ... | 3 | partially_solved | null | null | 3 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
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"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,397 | GRAPH-010 | Gyárfás-Sumner Conjecture | Is every graph class defined by excluding one fixed tree as an induced subgraph χ-bounded? | A graph class is χ-bounded if there's a function f such that every graph in the class with clique number ω has chromatic number at most f(ω). The conjecture states that forbidding any tree as an induced subgraph creates a χ-bounded class. This would unify many results on perfect graphs and their generalizations. The co... | 4 | partially_solved | null | null | 3 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,398 | GRAPH-011 | Jaeger's Petersen Coloring Conjecture | Does every bridgeless cubic graph have a cycle-continuous mapping to the Petersen graph? | Jaeger conjectured that every bridgeless cubic graph admits a special kind of homomorphism to the Petersen graph that preserves cycle structure. The Petersen graph plays a central role in graph theory as a universal counterexample and fundamental object. This conjecture connects graph homomorphisms, snarks (cubic graph... | 4 | partially_solved | null | null | 3 | null | 156 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,399 | GRAPH-012 | List Coloring Conjecture | For every graph, does the list chromatic index equal the chromatic index? | The chromatic index χ'(G) is the minimum number of colors needed to color edges so no two adjacent edges share a color. The list chromatic index is the minimum k such that edges can be colored from arbitrary k-element color lists. The conjecture states these are always equal. While proven for bipartite graphs and some ... | 4 | partially_solved | null | null | 3 | null | 198 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,400 | GRAPH-013 | Overfull Conjecture | Is a graph with maximum degree Δ(G) ≥ n/3 in class 2 if and only if it has an overfull subgraph with the same maximum degree? | By Vizing's theorem, every graph has chromatic index Δ or Δ+1 (class 1 or 2). A graph is overfull if it has more than Δ⌊n/2⌋ edges, forcing class 2. The overfull conjecture provides a complete characterization: when Δ ≥ n/3, being class 2 is equivalent to having an overfull subgraph preserving the maximum degree. This ... | 4 | partially_solved | null | null | 3 | null | 167 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"level": 4,
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"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,401 | GRAPH-014 | Total Coloring Conjecture | Is the total chromatic number of every graph at most Δ + 2, where Δ is the maximum degree? | Total coloring requires coloring both vertices and edges so adjacent/incident elements have different colors. Behzad and Vizing independently conjectured that the total chromatic number χ″(G) ≤ Δ(G) + 2. The lower bound Δ + 1 is easy (color each vertex and its incident edges distinctly). The upper bound Δ + 2 is proven... | 4 | partially_solved | null | null | 3 | null | 245 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
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"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,402 | GRAPH-015 | Albertson Conjecture | Can the crossing number of a graph be lower-bounded by the crossing number of a complete graph with the same chromatic number? | The crossing number is the minimum number of edge crossings in a planar drawing. Albertson conjectured cr(G) ≥ cr(K_χ(G)) where χ(G) is the chromatic number. This would link two fundamental graph parameters—crossing number and chromatic number. Proven for chromatic numbers up to 16, but the general case remains open. T... | 4 | partially_solved | null | null | 3 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,403 | GRAPH-016 | Conway's Thrackle Conjecture | Does every thrackle have at most as many edges as vertices? | A thrackle is a graph drawing where every pair of edges either meets at a common vertex or crosses exactly once. Conway conjectured that thrackles satisfy |E| ≤ |V|. Despite looking simple, this conjecture has resisted proof for decades. The best known bound is |E| ≤ 3|V|/2. This problem connects graph drawing with com... | 3 | partially_solved | null | null | 3 | null | 201 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,404 | GRAPH-017 | GNRS Conjecture | Do minor-closed graph families have $\ell_1$ embeddings with bounded distortion? | The GNRS conjecture asks whether graphs from minor-closed families (like planar graphs) can be embedded into L₁ space (ℓ₁ metric) with distortion bounded by a function of the excluded minor size. This connects graph theory with metric geometry and theoretical computer science. The conjecture has important implications ... | 5 | partially_solved | null | null | 3 | null | 145 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,405 | GRAPH-018 | Harborth's Conjecture | Can every planar graph be drawn with integer edge lengths? | Harborth conjectured that every planar graph has a straight-line drawing where all edge lengths are integers. While planar graphs always have straight-line drawings (Fáry's theorem), forcing integer lengths is much harder. Known for trees and some other classes, but open in general. This problem connects graph drawing ... | 4 | partially_solved | null | null | 3 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,406 | GRAPH-019 | Negami's Conjecture | Does every graph with a planar cover have a projective-plane embedding? | Negami conjectured that if a graph G has a planar cover (a planar graph that maps onto G), then G embeds in the projective plane. This would characterize projective-plane graphs via covering spaces. The conjecture connects topological graph theory with covering space theory from topology. Despite progress on special ca... | 4 | partially_solved | null | null | 3 | null | 156 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,407 | GRAPH-020 | Turán's Brick Factory Problem | What is the minimum crossing number of the complete bipartite graph $K_{m,n}$? | Turán's brick factory problem asks for the exact crossing number of complete bipartite graphs K_{m,n}. Zarankiewicz conjectured a formula in 1954, which is known to be correct for several cases but unproven in general. The problem arose from Turán observing workers crossing paths while moving bricks. Despite being simp... | 4 | partially_solved | null | null | 3 | null | 212 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,408 | GRAPH-021 | Guy's Crossing Number Conjecture | Is the crossing number of the complete graph $K_n$ equal to the value given by Guy's formula? | Guy conjectured a formula for the crossing number of complete graphs: cr(K_n) = (1/4)⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋. This is proven for n ≤ 12, but the general case is open. Finding the exact crossing number of complete graphs is a fundamental problem in topological graph theory. The conjecture represents our best gu... | 4 | partially_solved | null | null | 3 | null | 198 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,409 | GRAPH-022 | Universal Point Sets | Do planar graphs have universal point sets of subquadratic size? | A universal point set for n-vertex planar graphs is a set of points such that every n-vertex planar graph has a straight-line embedding on these points. Trivially, O(n²) points suffice. The question asks if o(n²) is possible. Best known lower bound is Ω(n), upper bound is O(n²). Closing this gap would advance our under... | 4 | partially_solved | null | null | 3 | null | 167 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,410 | GRAPH-023 | Conference Graph Existence | Does there exist a conference graph for every number of vertices $v > 1$ where $v \equiv 1 \pmod{4}$ and v is an odd sum of two squares? | A conference graph is a strongly regular graph with specific parameters related to conference matrices. The existence question for these graphs connects graph theory with number theory (sums of squares) and design theory. Known to exist for many values, but a complete characterization remains elusive. These graphs have... | 4 | partially_solved | null | null | 3 | null | 145 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,411 | GRAPH-024 | Conway's 99-Graph Problem | Does there exist a strongly regular graph with parameters (99,14,1,2)? | Conway asked whether a strongly regular graph with these specific parameters exists. The parameters pass all known necessary conditions (feasibility, integrality), but no construction is known. This is the smallest open case for strongly regular graphs. Finding such a graph or proving nonexistence would advance our und... | 4 | partially_solved | null | null | 3 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,412 | GRAPH-025 | Degree Diameter Problem | For given maximum degree d and diameter k, what is the largest possible number of vertices in a graph? | The degree diameter problem asks for the maximum order (number of vertices) of a graph with maximum degree d and diameter k. The Moore bound provides an upper limit, but it's rarely achieved (only for very special parameters). Finding the exact values or better bounds is a central problem in extremal graph theory with ... | 4 | partially_solved | null | null | 3 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,413 | GRAPH-026 | Moore Graph Existence | Does a Moore graph with girth 5 and degree 57 exist? | Moore graphs are extremal graphs achieving the Moore bound—the maximum possible vertices for given degree and diameter. The Hoffman-Singleton theorem shows Moore graphs with girth 5 can only have degree 2, 3, 7, or possibly 57. Graphs for degrees 2, 3, 7 are known (cycle C₅, Petersen, Hoffman-Singleton). Whether a degr... | 5 | partially_solved | null | null | 3 | null | 223 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,414 | GRAPH-027 | Barnette's Conjecture | Does every cubic bipartite three-connected planar graph have a Hamiltonian cycle? | Barnette's conjecture proposes that a specific family of planar graphs—cubic (3-regular), bipartite, and 3-connected—always contains Hamiltonian cycles. This strengthens Tait's conjecture (disproven by counterexamples) by adding bipartiteness. Despite extensive computational verification and many partial results, no pr... | 4 | partially_solved | null | null | 3 | null | 212 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,415 | GRAPH-028 | Chvátal's Toughness Conjecture | Is there a constant t such that every t-tough graph is Hamiltonian? | A graph is t-tough if removing any set S of vertices leaves at most |S|/t components. Chvátal conjectured that sufficiently tough graphs are Hamiltonian. Best known: every 2-tough graph on at least 3 vertices is Hamiltonian. But whether some finite t suffices in general is unknown. Toughness measures graph robustness; ... | 4 | partially_solved | null | null | 3 | null | 178 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,416 | GRAPH-029 | Cycle Double Cover Conjecture | Does every bridgeless graph have a collection of cycles that covers each edge exactly twice? | The cycle double cover conjecture asserts that every bridgeless graph has a family of cycles where each edge appears in exactly two cycles. Equivalent formulations involve graph embeddings and flows. Despite being open since the 1970s, this elegant conjecture connects cycle structure, graph embeddings, and topological ... | 4 | solved | null | null | 3 | null | 198 | 15 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | A proof of the cycle double cover conjecture was announced in July 2026. Sang-il Oum’s exposition presents the proof for every bridgeless graph. This is a reported resolution in an unrefereed preprint, not a claim of completed peer review.
**What remains.** Independent expert verification and publication of the announ... | [
{
"url": "https://arxiv.org/abs/2607.16356",
"label": "Sang-il Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition (2026)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates ... | 2026-09-27T00:00:00 | Sang-il Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition (2026): https://arxiv.org/abs/2607.16356 | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "unrefereed_preprint"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,417 | GRAPH-030 | Erdős-Gyárfás Conjecture | Does every graph with minimum degree 3 contain cycles of lengths that are powers of 2? | Erdős and Gyárfás conjectured that cubic graphs (minimum degree 3) must contain cycles whose lengths are all distinct powers of 2. The best known result is that such graphs contain cycles of Ω(log log n) distinct even lengths. This problem connects extremal graph theory with additive combinatorics and the structure of ... | 4 | partially_solved | null | null | 3 | null | 167 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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