id int64 1 30M | problem_number stringlengths 4 34 | title stringlengths 8 149 | statement stringlengths 11 35.2k | background stringlengths 395 52.6k | difficulty_level_id int64 1 5 | status stringclasses 3
values | proposed_by stringclasses 310
values | proposed_year int64 1.74k 2.03k ⌀ | category_id int64 1 20 | set_id int64 1 15 ⌀ | view_count int64 0 2.34k | favorite_count int64 0 156 | created_at timestamp[s]date 2024-01-01 00:00:00 2026-08-25 00:00:00 | updated_at timestamp[s]date 2024-01-01 00:00:00 2026-09-28 00:00:00 | published bool 2
classes | category dict | difficulty dict | set dict | literature_assessment stringlengths 80 804 ⌀ | literature_sources listlengths 1 6 ⌀ | literature_checked_at timestamp[s]date 2026-08-21 00:00:00 2026-09-28 00:00:00 ⌀ | solution_reference stringclasses 257
values | status_review dict | tags listlengths 1 5 ⌀ | clean_statement stringlengths 11 35.2k ⌀ | original_statement stringlengths 11 35.7k ⌀ | statement_status stringclasses 4
values | statement_verification stringlengths 5 900 ⌀ | archive_reason stringclasses 26
values | research_classification stringclasses 4
values | research_summary stringlengths 40 1.95k ⌀ | research_difficulty_suggested stringclasses 14
values | historical_research_summary stringclasses 22
values | research_summary_reviewed_at timestamp[s]date 2026-09-28 00:00:00 2026-09-28 00:00:00 ⌀ | source_url stringlengths 25 84 ⌀ | source_citation stringlengths 75 207 ⌀ | duplicate_of listlengths 0 2 ⌀ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1,864 | GUY-A10 | Gilbreath's Conjecture | Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it true that $d_1^k = 1$ for all $k$? | Gilbreath conjectured this (and Proth claimed to have proved it long before). This was verified for $k < 63419$ by Killgrove & Ralston. Odlyzko checked it for primes up to $\pi(10^{13})$. Croft and others suggest it has nothing to do with primes as such, but will be true for any sequence consisting of 2 and odd numbers... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,865 | GUY-A11 | Erdős $100 Problem on Increasing and Decreasing Gaps | Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - p_n$? | Erdős & Turán showed that the values of $n$ for which $d_n > d_{n+1}$ have positive lower density, but it is not known if there are infinitely many increasing or decreasing sets of three consecutive values of $d_n$. Erdős offers $100.00 for a proof that such an $n_0$ does not exist. From Richard Guy's "Unsolved Problem... | 3 | solved | null | null | 1 | 9 | 0 | 0 | 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": 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 | Reconciled with the existing literature review of 2026-08-17: The proposed eventual strict alternation of consecutive prime gaps is false: arbitrarily long strictly increasing and arbitrarily long strictly decreasing runs of consecutive prime gaps exist. This is a consistency correction based on that cited review; the ... | [
{
"url": "https://doi.org/10.4064/aa167-3-4",
"label": "William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arithmetica 167 (2015), 261-266."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "Unsolve... | 2026-09-27T00:00:00 | William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arithmetica 167 (2015), 261-266.: https://doi.org/10.4064/aa167-3-4 | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "triage_reconciliation",
"evidence": "existing_literature_review"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,866 | GUY-A13 | Erdős Conjecture on Carmichael Numbers | Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity? | Erdős conjectured this behavior for the count of Carmichael numbers. Alford, Granville & Pomerance showed there are infinitely many Carmichael numbers, in fact more than $x^\beta$ of them less than $x$ for $\beta > 0.290306$. Pomerance, Selfridge & Wagstaff give a heuristic argument supporting Erdős' conjecture. From R... | 4 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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 |
1,867 | GUY-A14a | Pomerance's Questions on Good Primes | Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \le i \le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0? Are there infinitely many $n$ with $p_n p_{n+1} > p_{n-i} p_{n+1+i}$ for all $i$, $1 \le i \le n-1$? | Erdős and Straus introduced the concept of good primes. Examples include 5, 11, 17, and 29. Pomerance used the prime number graph to show there are infinitely many good primes and posed several related questions. From Richard Guy's "Unsolved Problems in Number Theory", Section A14.
<!-- LITERATURE-TRIAGE:BEGIN -->
## ... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,868 | GUY-A15 | Congruent Products of Consecutive Numbers | What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\prod_{i=1}^{k_1} (a+i) \equiv \prod_{i=1}^{k_2} (a+k_1+i) \equiv \prod_{i=1}^{k_3} (a+k_1+k_2+i) \equiv 1 \pmod{p}$? | Erdős observed that $3 \cdot 4 \equiv 5 \cdot 6 \cdot 7 \equiv 1 \pmod{11}$ and suggested that such primes $p$ exist for any number of congruent products. Narkiewicz and others found examples for larger numbers of terms. From Richard Guy's "Unsolved Problems in Number Theory", Section A15.
<!-- LITERATURE-TRIAGE:BEGIN... | 2 | solved | null | null | 1 | 9 | 0 | 0 | 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": 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 | Reconciled with the existing literature review of 2026-08-17: The least prime is 17. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claims.
**What remains.** Nothing remains for the stated least-prime question; analogous least primes for larger nu... | [
{
"url": "https://oeis.org/A060427",
"label": "OEIS Foundation, A060427, Smallest prime p such that there are n strings of consecutive integers all having products = 1 mod p, updated 2026-07-27."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedM... | 2026-09-27T00:00:00 | OEIS Foundation, A060427, Smallest prime p such that there are n strings of consecutive integers all having products = 1 mod p, updated 2026-07-27.: https://oeis.org/A060427 | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "triage_reconciliation",
"evidence": "existing_literature_review"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,869 | GUY-A16 | Walking to Infinity on Gaussian Primes | Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length? | Motzkin and Gordon asked this question about Gaussian primes (primes in the ring of complex numbers $a+bi$ where $a, b$ are integers). Presumably not. Jordan & Rabung showed that steps of length at least 4 are necessary. Gethner, Wagon & Wick produced a moat of width $\sqrt{26}$. From Richard Guy's "Unsolved Problems i... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,870 | GUY-A17 | Giuga's Conjecture on Prime Characterization | Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \dots + (n-1)^{n-1} + 1$, then $n$ is prime? | Sierpiński observed that if $n$ is prime, then $n$ divides this sum. Giuga conjectured the converse and verified it for $n \le 10^{1000}$. A counterexample would be a Carmichael number with additional properties. An equivalent conjecture is $n B_{n-1} \equiv -1 \pmod{n}$ where $B_k$ are Bernoulli numbers. From Richard ... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,871 | GUY-A18 | Erdős-Selfridge Classification: Infinitely Many Primes in Each Class | In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of $p+1$ are 2 or 3; and $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\le r-1$, with equality for at least one prime factor. | The first few classes are: Class 1: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, ...; Class 2: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, ...; Class 3: 37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, ... From Richard Guy's "Unsolved Problems in Number Theory", Section A18.
<!-- LITERATURE-TRI... | 3 | open | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,872 | GUY-A19a | Erdős Conjecture on $n - 2^k$ Prime | Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \le 2^k < n$? | Erdős conjectures that these are the only such values. He also conjectures that for infinitely many $n$, all the integers $n - 2^k, 1 \le 2^k < n$ are squarefree. From Richard Guy's "Unsolved Problems in Number Theory", Section A19.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:*... | 3 | open | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,873 | GUY-A19b | Cohen-Selfridge Problem on $\pm p^a \pm 2^b$ | What is the least positive odd number not of the form $\pm p^a \pm 2^b$, where $p$ is an odd prime? | Cohen & Selfridge observed that the number is greater than $2^{18}$. This is related to the representation of odd numbers as sums or differences of prime powers and powers of 2. From Richard Guy's "Unsolved Problems in Number Theory", Section A19.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-... | 2 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,874 | GUY-A20 | Density of Symmetric Primes | Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the diagonal. A pair is symmetric if $S(p, q) = S(q,p)$. Is the number of symmetric primes less than $x$ equal to $x/(\ln x)^{\sigma+o(1)}$, where $\sigma = 2 - (1+\ln \ln 2)/\l... | Fletcher, Lindgren & Pomerance showed that a pair is symmetric just if $|p - q| = (p - 1, q - 1)$, and that the number of symmetric primes less than $x$ is at most $x/(\ln x)^{1.027}$. They conjectured the more precise asymptotic. From Richard Guy's "Unsolved Problems in Number Theory", Section A20.
<!-- LITERATURE-TR... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,875 | GUY-A12a | Square Pseudoprimes | Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$? | Pinch observed that there are 54 non-squarefree pseudoprimes up to $10^{13}$, all multiples of $1093^2$ or $3511^2$. The question asks if there are other perfect squares that are pseudoprimes to base 2. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,876 | GUY-A12b | Selfridge-Wagstaff-Pomerance Prize Problem | Does there exist a composite number $n \equiv 3$ or $7 \pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$? | Selfridge, Wagstaff & Pomerance offer $500 + $100 + $20 = $620 for finding such a composite $n$, or $20 + $100 + $500 = $620 for a proof that no such $n$ exists. This combines pseudoprime properties with Fibonacci divisibility. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
<!-- LITERATURE-TRIAG... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,877 | GUY-A12c | Even Fibonacci Pseudoprimes | Does there exist an even Fibonacci pseudoprime? | A Fibonacci pseudoprime of the $m$-th kind is an odd composite integer $n$ with $V_n(m, -1) \equiv m \pmod n$ where $V_n$ is the Lucas sequence. Somer showed that if an even Fibonacci pseudoprime exists, it must be greater than $28 \times 10^{12}$. From Richard Guy's "Unsolved Problems in Number Theory", Section A12.
... | 3 | partially_solved | null | null | 1 | 9 | 0 | 0 | 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": 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 |
1,878 | EP-1 | Erdős Problem #1 | If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $ N \gg 2^{n}. $ | Erd\H{o}s called this 'perhaps my first serious problem' (in \cite{Er98} he dates it to 1931). The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erd\H{o}s and Moser \cite{Er56} proved $ N\geq (\tfrac{1}{... | 3 | solved | null | null | 1 | 8 | 0 | 0 | 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": 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"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 1 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1",
"label": "Erdős Problems #1: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hugging... | 2026-09-27T00:00:00 | Erdős Problems #1: disproved (Lean): https://www.erdosproblems.com/1
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,879 | EP-3 | Erdős Problem #3 | If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions? | This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like $ r_k(N) \ll_k \frac{N}{(\log N)(\log\log N)^2} $ would be sufficient.
Even the case $k=3$ is non-trivial, but was proved by Bloom a... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,880 | EP-5 | Erdős Problem #5 | Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $ \lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C? $ | Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:
{UL}
{LI}$\infty\in S$ by Westzynthius' result \cite{We31} on large prime gaps,{/LI}
{LI}$0\in S$ by the work of Goldston, Pintz, ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,881 | EP-9 | Erdős Problem #9 | Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive? | In \cite{Er77c} Erd\H{o}s credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. Crocker \cite{Cr71} has proved there are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$. Pan \cite{Pa11} improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. E... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,882 | EP-10 | Erdős Problem #10 | Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2? | Erd\H{o}s described this as 'probably unattackable'. In \cite{ErGr80} Erd\H{o}s and Graham suggest that no such $k$ exists. Gallagher \cite{Ga75} has shown that for any $\epsilon>0$ there exists $k(\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\epsilon)$ many powers of 2 has lower... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,883 | EP-12 | Erdős Problem #12 | Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0? $ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $ \lvert A\cap\{1,\ldot... | Asked by Erd\H{o}s and S\'{a}rk"{o}zy \cite{ErSa70}, who proved that $A$ must have density $0$. They also prove that this is essentially best possible, in that given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with this property and infinitely many $N$ such that $ \lvert A\cap\{1,\ldots,N\}\rv... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,884 | EP-14 | Erdős Problem #14 | Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.
Is it true that for all $\epsilon>0$ and large $N$ $ \lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}? $ Is it possible that $ \lvert \{1,\... | Apparently originally considered by Erd\H{o}s and Nathanson, although later Erd\H{o}s attributes this to Erd\H{o}s, S\'{a}rk"{o}zy, and Szemer\'{e}di (but gives no reference), and claims a construction of an $A$ such that for all $\epsilon>0$ and all large $N$ $ \lvert \{1,\ldots,N\}\backslash B\rvert \ll_\epsilon N^{1... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,885 | EP-15 | Erdős Problem #15 | Is it true that $ \sum_{n=1}^\infty(-1)^n\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes? | Erd\H{o}s suggested that a computer could be used to explore this, and did not see any other method to attack this.
Tao \cite{Ta23} has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.
In \cite{Er98} Erd\H{o}s further conjectures that $ \sum_{n=1}^\infty (-1)... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,886 | EP-17 | Erdős Problem #17 | Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$? | The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.
Blecksmith, Erd\H{o}s, and Selfridge \cite{BES99} proved that the number of such primes is $ \ll_A \frac{x}{(\log x)^A} $ for every $A>0$, and Elsholtz \cite{El03} improved this to $ \ll x... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,887 | EP-18 | Erdős Problem #18 | We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.
Are there infinitely many practical $m$ such that $ h(m) < (\log\log m)^{O(1)}? $ Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{... | It is easy to see that almost all numbers are not practical. Erd\H{o}s originally showed that $h(n!) <n$. Vose \cite{Vo85} proved the existence of infinitely many practical $m$ such that $h(m)\ll (\log m)^{1/2}$.
The sequence of practical numbers is A005153 in the OEIS.
The reward of \$250 is offered in \cite{Er81h}, a... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,888 | EP-20 | Erdős Problem #20 | Let $f(n,k)$ be minimal such that every family $\mathcal{F}$ of $n$-uniform sets with $\lvert \mathcal{F}\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that $ f(n,k) < c_k^n $ for some constant $c_k>0$? | Erd\H{o}s and Rado \cite{ErRa60} originally proved $f(n,k)\leq (k-1)^nn!$. Kostochka \cite{Ko97} improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erd\H{o}s awarded him the consolation prize of \$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,889 | EP-25 | Erdős Problem #25 | Let $n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n
ot\equiv a_i\pmod{n_i}$. Must the logarithmic density of $A$ exist? | This is a special case of [486].
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The one-residue-class-per-modulus logarithmic-density question remains open; it is a special case of EP-486.
**Verified pa... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,890 | EP-28 | Erdős Problem #28 | If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$. | Conjectured by Erd\H{o}s and Tur\'{a}n. They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log n>0$.
Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.
Erd\H{o}s and S\'{a}rk"{o}zy conjectured the stronger version that if $A=\... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,891 | EP-30 | Erdős Problem #30 | Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$, $ h(N) = N^{1/2}+O_\epsilon(N^\epsilon)? $ | A problem of Erd\H{o}s and Tur\'{a}n. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erd\H{o}s remarks this is perhaps too optimistic. Erd\H{o}s and Tur\'{a}n \cite{ErTu41} proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindstr"{o}m \cite{Li69}. Both proofs in fact give $ h(N) \leq N^{1/... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,892 | EP-32 | Erdős Problem #32 | Is there a set $A\subset\mathbb{N}$ such that $ \lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2) $ and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$?
Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1? $ | Such a set is called an additive complement to the primes.
Erd\H{o}s \cite{Er54} proved that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of Lorentz \cite{Lo54} who achieved $\ll (\log N)^3$).
Wolke \cite{Wo96} has shown that such a bound is almost true, in th... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,893 | EP-33 | Erdős Problem #33 | Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible value of $ \limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}? $ Is $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1? $ | Such a set $A$ is called an additive complement of the set of squares. Erd\H{o}s observed that there exist $A$ for which the $\limsup$ is finite and $>1$. Moser \cite{Mo65} proved that, for any such $A$, $ \liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1.06. $ The best-known lower bound is $ \liminf \frac{\lv... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,894 | EP-36 | Erdős Problem #36 | Find the optimal constant $c>0$ such that the following holds.
For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\in A$ and $b\in B$ is at least $cN$. | The minimum overlap problem. The example (with $N$ even) $A=\{N/2+1,\ldots,3N/2\}$ shows that $c\leq 1/2$ (indeed, Erd\H{o}s initially conjectured that $c=1/2$). The lower bound of $c\geq 1/4$ is trivial, and Scherk improved this to $1-1/\sqrt{2}=0.29\cdots$. The current records are $ 0.379005 < c < 0.380924, $ the low... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,895 | EP-38 | Erdős Problem #38 | Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b\in B$ such that $ \lvert (A\cup (A+b))\cap \{1,\ldots,N\}\rvert\geq (\alpha+f(\alpha))N $ where $f(\alpha)>0$ for $0<\alpha <1 $?
The... | Erd\H{o}s \cite{Er36c} proved that if $B$ is an additive basis of order $k$ then, for any set $A$ of Schnirelmann density $\alpha$, for every $N$ there exists some integer $b\in B$ such that $ \lvert (A\cup (A+b))\cap \{1,\ldots,N\}\rvert\geq \left(\alpha+\frac{\alpha(1-\alpha)}{2k}\right)N. $ It seems an interesting q... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 38 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/38",
"label": "Erdős Problems #38: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingf... | 2026-09-27T00:00:00 | Erdős Problems #38: proved (Lean): https://www.erdosproblems.com/38
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,896 | EP-39 | Erdős Problem #39 | Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $ \lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon} $ for all $\epsilon>0$? | The trivial greedy construction achieves $\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Koml\'{o}s, and Szemer\'{e}di \cite{AKS81b}, who found an infinite Sidon set with growth rate $\gg (N\log N)^{1/3}$. The current best bound of $\gg N^{\sqrt{2}-1+o(1)}$ is due to Ruzsa \cite{Ru98}.
Erd\H{o}s \ci... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,897 | EP-40 | Erdős Problem #40 | For what functions $g(N)\to \infty$ is it true that $ \lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)} $ implies $\limsup 1_A\ast 1_A(n)=\infty$? | This is a stronger form of the Erd\H{o}s-Tur\'{a}n conjecture [28] (since establishing this for any function $g(N)\to \infty$ would imply a positive solution to [28]).
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature as... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,898 | EP-41 | Erdős Problem #41 | Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that $ \liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0? $ | Erd\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then $ \liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0. $ This is discussed in problem C11 of Guy's collection \cite{Gu04}, in which Guy says Erd\H{o}s offered \$500 for the general problem of whether, for ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,899 | EP-42 | Erdős Problem #42 | Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set $B\subset \{1,\ldots,N\}$ of size $M$ such that $(A-A)\cap(B-B)=\{0\}$? | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** The maintained Erdős Problems source lists problem 42 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**V... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 42 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/42",
"label": "Erdős Problems #42: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingf... | 2026-09-27T00:00:00 | Erdős Problems #42: solved (Lean): https://www.erdosproblems.com/42
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,900 | EP-43 | Erdős Problem #43 | If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that $ \binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1), $ where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound b... | Since it is known that $f(N)\sim \sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\lvert A\rvert=\lvert B\rvert$, $ \lvert A\rvert \leq \left(\frac{1}{\sqrt{2}}-c+o(1)\right)\sqrt{N} $ for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.
In th... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 43 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/43",
"label": "Erdős Problems #43: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggi... | 2026-09-27T00:00:00 | Erdős Problems #43: disproved (Lean): https://www.erdosproblems.com/43
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,901 | EP-44 | Erdős Problem #44 | Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$? | See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).
This is discussed in problem C9 of Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
<!... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,902 | EP-50 | Erdős Problem #50 | Schoenberg proved that for every $c\in [0,1]$ the density of $ \{ n\in \mathbb{N} : \phi(n)<cn\} $ exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive? | Erd\H{o}s \cite{Er95} could prove the distribution function is purely singular.
References
[Er95] Erd\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Cl... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,903 | EP-51 | Erdős Problem #51 | Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$? | Carmichael has asked whether there is an integer $t$ for which $\phi(n)=t$ has exactly one solution. Erd\H{o}s has proved that if such a $t$ exists then there must be infinitely many such $t$.
See also [694].
This is discussed in problems B36 and B39 of Guy's collection \cite{Gu04}.
References
[Gu04] Guy, Richard K.,... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,904 | EP-52 | Erdős Problem #52 | Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $ \max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}? $ | The sum-product problem. Erd\H{o}s and Szemer\'{e}di \cite{ErSz83} proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $ \lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right) $ for some constant $c>0$. The lower bound has been improved a number ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,905 | EP-60 | Erdős Problem #60 | Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$? | Conjectured by Erd\H{o}s and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.
The behaviour of $\mathrm{ex}(n;C_4)$ is the subject of [765].
He, Ma, and Yang \cite{HeMaYa21} have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.
References
[HeMaYa21] He, J. and M... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,906 | EP-61 | Erdős Problem #61 | For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\geq n^c$ vertices? | Conjectured by Erd\H{o}s and Hajnal \cite{ErHa89}, who proved that a complete graph or independent set must exist on $ \geq \exp(c_H\sqrt{\log n}) $ many vertices, where $c_H>0$ is some constant. This was improved by Buci\'{c}, Nguyen, Scott, and Seymour \cite{BNSS23} to $ \geq \exp(c_H\sqrt{\log n\log\log n}). $ See a... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,907 | EP-62 | Erdős Problem #62 | If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\aleph_0$) which is a subgraph of both $G_1$ and $G_2$? | Erd\H{o}s also asked \cite{Er87} about finding a common subgraph $H$ (with chromatic number either $4$ or $\aleph_0$) in any finite collection of graphs with chromatic number $\aleph_1$.
Every graph with chromatic number $\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594].... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,908 | EP-65 | Erdős Problem #65 | Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that $ \sum\frac{1}{a_i}\gg \log k? $ Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph? | A problem of Erd\H{o}s and Hajnal.
Gy\'{a}rf\'{a}s, Koml\'{o}s, and Szemer\'{e}di \cite{GKS84} have proved that this sum is $\gg \log k$, so that only the second question remains. Liu and Montgomery \cite{LiMo20} have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.
See also the entry in... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,909 | EP-66 | Erdős Problem #66 | Is there $A\subseteq \mathbb{N}$ such that $ \lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n} $ exists and is $
eq 0$? | A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erd\H{o}s believed the answer should be no. Erd\H{o}s and S\'{a}rk"{o}zy proved that $ \frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,910 | EP-68 | Erdős Problem #68 | Is $ \sum_{n\geq 2}\frac{1}{n!-1} $ irrational? | The decimal expansion is A331373 in the OEIS. Weisenberg has observed that this sum can also be written as $ \sum_{k\geq 1}\sum_{n\geq 2}\frac{1}{(n!)^k}. $ Erd\H{o}s \cite{Er88c} notes that $\sum \frac{1}{n!+t}$ should be transcendental for every integer $t$.
References
[Er88c] Erd"{o}s, P., On the irrationality of ... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,911 | EP-70 | Erdős Problem #70 | Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta, n)_2^3$? | Erd\H{o}s and Rado proved that $\mathfrak{c}\to (\omega+n,4)_2^3$ for any $2\leq n<\omega$.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The full triple partition relation for arbitrary countable beta ... | 1 | open | null | null | 10 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,912 | EP-74 | Erdős Problem #74 | Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges? | Conjectured by Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}.
R"{o}dl \cite{Ro82} has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.
It is open even for $f(n)=\sqrt{n}$. Erd\H{o}s offered \$500 for a pro... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 74 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/74",
"label": "Erdős Problems #74: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggi... | 2026-09-27T00:00:00 | Erdős Problems #74: disproved (Lean): https://www.erdosproblems.com/74
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,913 | EP-75 | Erdős Problem #75 | Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\epsilon}$? | Conjectured by Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}. In \cite{Er95d} Erd\H{o}s suggests this may even be true with an independent set of size $\gg n$.
See also [750].
References
[EHS82] Erd\H{o}s, P. and Hajnal, A. and Szemer\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of ... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,914 | EP-77 | Erdős Problem #77 | If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of $ \lim_{k\to \infty}R(k)^{1/k}. $ | Erd\H{o}s offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In \cite{Er88} he raises this prize to \$10000). Erd\H{o}s proved $ \sqrt{2}\leq \limin... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,915 | EP-78 | Erdős Problem #78 | Give a constructive proof that $R(k)>C^k$ for some constant $C>1$. | Erd\H{o}s gave a simple probabilistic proof that $R(k) \gg k2^{k/2}$.
Equivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\geq c\log n$ for some constant $c>0$.
In \cite{Er69b} Erd\H{o}s asks for even a construction whose... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,916 | EP-80 | Erdős Problem #80 | Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles.
Estimate $f_c(n)$. In particular, is it true that $f_c... | A problem of Erd\H{o}s and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\ll_c n^{1/2}$. Szemer\'{e}di observed that his regularity lemma implies that $f_c(n)\to \infty$.
Edwards (unpublished) and Khadziivanov and Nikiforov \cite{KhNi79} proved independently that $f_c(n) \geq n/6$ when $c>1/4$ (see... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,917 | EP-81 | Erdős Problem #81 | Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques? | Asked by Erd\H{o}s, Ordman, and Zalcstein \cite{EOZ93}, who proved an upper bound of $(1/4-\epsilon)n^2$ many cliques (for some very small $\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.
A split graph... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,918 | EP-82 | Erdős Problem #82 | Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log n\to \infty$. | Conjectured by Erd\H{o}s, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$.
Ramsey's theorem implies that $F(n)\gg \log n$. Bollob\'{a}s observed that $F(n)\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov \cite{AKS07} have improved this to $n^{1/2}(\log n)^{O(1)}$.
In \cite{Er93} Erd\H{o}s asks whether,... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,919 | EP-84 | Erdős Problem #84 | The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$.
Prove that $f(n)=o(2^n)$.
Prove that $f(n)/2^{n/2}\to \infty$. | Conjectured by Erd\H{o}s and Faudree, who showed that $2^{n/2}<f(n) \leq 2^{n-2}$. The first problem was solved by Verstra"{e}te \cite{Ve04}, who proved $ f(n)\ll 2^{n-n^{1/10}}. $ This was improved by Nenadov \cite{Ne25} to $ f(n) \ll 2^{n-n^{1/2-o(1)}}. $ One can also ask about the existence and value of $\lim f(n)^{... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,920 | EP-86 | Erdős Problem #86 | Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with $ \geq \left(\frac{1}{2}+o(1)\right)n2^{n-1} $ many edges contains a $C_4$? | Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\leq (\frac{1}{2}+o(1))n2^{n-1}$.
Erd\H{o}s \cite{Er91} showed that $ f(n) \geq \left(\frac{1}{2}+\frac{c}{n}\right)n2^{n-1} $ for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determi... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,921 | EP-87 | Erdős Problem #87 | Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $ R(G)>(1-\epsilon)^kR(k) $ for every graph $G$ with chromatic number $\chi(G)=k$?
Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$? | Erd\H{o}s originally conjectured that $R(G)\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay \cite{FaMc93} showed that $R(W)=17$ for the pentagonal wheel $W$.
Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 3/4$. Yuval Wigderson points out that $R(G)\gg 2^{k/2}$ for any $G... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,922 | EP-89 | Erdős Problem #89 | Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances? | A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \cite{GuKa15} who proved that there are always $\gg n/\log n$ many distinct distances.
A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct dista... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,923 | EP-90 | Erdős Problem #90 | Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart? | The unit distance problem. In \cite{Er94b} Erd\H{o}s dates this conjecture to 1946. In \cite{Er82e} he offers \$300 for the upper bound $n^{1+o(1)}$.
This would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 90 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/90",
"label": "Erdős Problems #90: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggi... | 2026-09-27T00:00:00 | Erdős Problems #90: disproved (Lean): https://www.erdosproblems.com/90
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,924 | EP-91 | Erdős Problem #91 | Let $n$ be a sufficently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar. | For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.
For $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erd\H{o}s mysteriously remarks in \cite{Er90} this was proved by 'a colleagu... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,925 | EP-92 | Erdős Problem #92 | Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.
Is it true that $f(n)\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\log\log n)}$? | This is a stronger form of the unit distance conjecture (see [90]).
The set of lattice points imply $f(n) > n^{c/\log\log n}$ for some constant $c>0$. Erd\H{o}s offered \$500 for a proof that $f(n) \leq n^{o(1)}$ but only \$100 for a counterexample. This latter prize is downgraded to \$50 in \cite{ErFi97}.
It is trivia... | 1 | solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 92 as “disproved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/92",
"label": "Erdős Problems #92: disproved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface.... | 2026-09-27T00:00:00 | Erdős Problems #92: disproved: https://www.erdosproblems.com/92
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,926 | EP-96 | Erdős Problem #96 | If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart. | Conjectured by Erd\H{o}s and Moser. In \cite{Er92e} Erd\H{o}s credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. F"{u}redi \cite{Fu90} proved an upper bound of $O(n\log n)$. A short proof of this bound was given by Brass and Pach \cite{BrPa01}. The best known upper bound is $ \leq n\log_2... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,927 | EP-98 | Erdős Problem #98 | Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$? | Erd\H{o}s could not even prove $h(n)\geq n$. Pach has shown $h(n)<n^{\log_23}$. Erd\H{o}s, F"{u}redi, and Pach \cite{EFPR93} have improved this to $ h(n) < n\exp(c\sqrt{\log n}) $ for some constant $c>0$.
References
[EFPR93] Erd\H{o}s, Paul and F"{u}redi, Zolt\'{a}n and Pach, J\'{a}nos and
Ruzsa, Imre Z., The grid re... | 1 | open | null | null | 6 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,928 | EP-99 | Erdős Problem #99 | Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1? | Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erd\H{o}s believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).
Erd\H{o}s \cite{Er94b} wrote 'I could not prov... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,929 | EP-100 | Erdős Problem #100 | Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$? | Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz \cite{GuKa15} imply a lower bound of $\gg n/\log n$.
References
[Gu... | 1 | open | null | null | 6 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,930 | EP-101 | Erdős Problem #101 | Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$. | There are examples of sets of $n$ points with $\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Gr"{u}nbaum, and Sloane \cite{BGS74} and F"{u}redi and Pal\'{a}sti \cite{FuPa84}.
Gr"{u}nbaum \cite{Gr76} constructed an example with $\gg n^{3/2}$ such lines. Erd\H{o}s ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,931 | EP-102 | Erdős Problem #102 | Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\to \infty$? | A problem of Erd\H{o}s and Purdy. It is not even known if $h_c(n)\geq 5$ (see [101]).
It is easy to see that $h_c(n) \ll_c n^{1/2}$, and Erd\H{o}s at one point \cite{Er95} suggested that perhaps a similar lower bound $h_c(n)\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,932 | EP-103 | Erdős Problem #103 | Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x
eq y$. Is it true that $h(n)\to \infty$? | It is not even known whether $h(n)\geq 2$ for all large $n$.
See also [99].",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The number of incongruent diameter-minimizing unit-... | 1 | open | null | null | 6 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,933 | EP-104 | Erdős Problem #104 | Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$. | In \cite{Er81d} Erd\H{o}s proved that $\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erd\H{o}s claims in a number of places this produce... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,934 | EP-108 | Erdős Problem #108 | For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth $\geq r$ and chromatic number $\geq k$? | Conjectured by Erd\H{o}s and Hajnal. R"{o}dl \cite{Ro77} has proved the $r=4$ case (see [923]). The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.
In \cite{Er79b} Erd\H{o}s also asks whether $ \lim_{k\to \infty}\frac... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,935 | EP-111 | Erdős Problem #111 | If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.
What is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\to \infty$ for every graph $G$ with chromatic number $\aleph_1$? | A problem of Erd\H{o}s, Hajnal, and Szemer\'{e}di \cite{EHS82}. Every $G$ with chromatic number $\aleph_1$ must have $h_G(n)\gg n$ since $G$ must contain, for some $r$, $\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.
On the other hand, Erd\H{o}s, Hajnal, and Szemer\'{e}di proved that there is a $G$ with ch... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,936 | EP-112 | Erdős Problem #112 | Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$. | A problem of Erd\H{o}s and Rado \cite{ErRa67}, who showed $k(n,m) \ll_m n^{m-1}$, or more precisely, $ k(n,m) \leq \frac{2^{m-1}(n-1)^m+n-2}{2n-3}. $ Larson and Mitchell \cite{LaMi97} improved the dependence on $m$, establishing in particular that $k(n,3)\leq n^{2}$. Zach Hunter has observed that $ R(n,m) \leq k(n,m)\l... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,937 | EP-114 | Erdős Problem #114 | If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$? | A problem of Erd\H{o}s, Herzog, and Piranian \cite{EHP58}. It is also listed as Problem 4.10 in \cite{Ha74}, where it is attributed to Erd\H{o}s.
Let the maximal length of such a curve be denoted by $f(n)$.
{UL}
{LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular t... | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,938 | EP-117 | Erdős Problem #117 | Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x
eq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.
Estimate $h(n)$ as well as possible. | Pyber \cite{Py87} has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$. Erd\H{o}s \cite{Er97f} writes that the lower bound was already known to Isaacs.
References
[Er97f] Erd\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[Py87] Pyber... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,939 | EP-119 | Erdős Problem #119 | Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let $ p_n(z)=\prod_{i\leq n} (z-z_i). $ Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$.
Is it true that $\limsup M_n=\infty$?
Is it true that there exists $c>0$ such that for infinitely many $n... | This is Problem 4.1 in \cite{Ha74} where it is attributed to Erd\H{o}s.
The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner \cite{Wa80}, who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.
The second question was answered by Beck \cite{Be91}, who proved that there exists some $c>0... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 119 as “solved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/119",
"label": "Erdős Problems #119: solved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #119: solved (Lean): https://www.erdosproblems.com/119
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,940 | EP-120 | Erdős Problem #120 | Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a
eq 0$? | The Erd\H{o}s similarity problem.
This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.
Steinhaus \cite{St20} has proved this is false whenever $A$ is a finite set.
This conjecture is ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,941 | EP-122 | Erdős Problem #122 | For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinitely many $x$ such that $ \frac{\#\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty? $ | Asked by Erd\H{o}s, Pomerance, and S\'{a}rk"{o}zy \cite{EPS97} who prove that this is true when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erd\H{o}s believed it is false when $f(n)=\phi(n)$ or $\sigma(n)$.
References
[EPS97] Erd\H{o}s, Paul and Pomerance, Carl and S\'{a}rk"{o}zy,... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,942 | EP-123 | Erdős Problem #123 | Let $a,b,c\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq 0$), none of which divide any other? | A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erd\H{o}s and Lewin \cite{ErLe96}, who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.
As a p... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 123 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/123",
"label": "Erdős Problems #123: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #123: proved (Lean): https://www.erdosproblems.com/123
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,943 | EP-124 | Erdős Problem #124 | For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that $ \sum_{1\leq i\leq r}\frac{1}{d_r-1}\geq 1. $ Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $... | The second question was conjectured by Burr, Erd\H{o}s, Graham, and Li \cite{BEGL96}, who proved it for $\{3,4,7\}$.
The first question was asked separately by Erd\H{o}s in \cite{Er97} and \cite{Er97e} (although there is some ambiguity over whether he intended $P(d,0)$ or $P(d,1)$ - certainly he mentions no gcd conditi... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,944 | EP-125 | Erdős Problem #125 | Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $4$.
Does $A+B$ have positive density? | A problem of Burr, Erd\H{o}s, Graham, and Li \cite{BEGL96}. More generally, if $n_1<\cdots<n_k$ have $ \sum_{i=1}^k\log_{n_k}(2)>1 $ and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\cdots+A_k$ have positive density? Melfi \cite{Me01} noted this is false as written, with a counte... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 125 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/125",
"label": "Erdős Problems #125: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hug... | 2026-09-27T00:00:00 | Erdős Problems #125: disproved (Lean): https://www.erdosproblems.com/125
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,945 | EP-126 | Erdős Problem #126 | Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a
eq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$? | Investigated by Erd\H{o}s and Tur\'{a}n \cite{ErTu34} (prompted by a question of L\'{a}z\'{a}r and Gr"{u}nwald) in their first joint paper, where they proved that $ \log n \ll f(n) \ll n/\log n $ (the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erd\H{o}s says that $f(n)=o(n/\log n)$ has never been proved, but p... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 126 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/126",
"label": "Erdős Problems #126: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #126: proved (Lean): https://www.erdosproblems.com/126
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,946 | EP-129 | Erdős Problem #129 | Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that $ R(n;3,r) < C^{\sqrt{n}}. $ | Conjectured by Erd\H{o}s and Gy\'{a}rf\'{a}s, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erd\H{o}s thought it likely that for all $r,k\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that $ C_1^{n^{1/k-1}}< R... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source still lists #129 as “open” on 2026-09-27. The August literature triage claimed a resolution, but the discussion explicitly flags that conclusion as unverified. That claim is not accepted as a full solution here.
**What remains.** Resolve the discrepancy between the earlier triage c... | [
{
"url": "https://www.erdosproblems.com/129",
"label": "Erdős Problems #129: open"
},
{
"url": "https://www.erdosproblems.com/129",
"label": "Thomas F. Bloom, Erdős Problem #129, with disproof attributed to Antonio Girão, accessed 2026-08-17."
},
{
"url": "https://www.erdosproblems.com/l... | 2026-09-27T00:00:00 | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "needs_review",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,947 | EP-130 | Erdős Problem #130 | Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.
How large can the chromatic number and clique number of t... | Asked by Andr\'{a}sfai and Erd\H{o}s. Erd\H{o}s \cite{Er97b} also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erd\H{o}s \cite{AnEr45}.
See also [213].
References
[AnEr45] Anning, Norman H. and Erd\H{o}s, Paul, Integral distances. Bull. A... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,948 | EP-131 | Erdős Problem #131 | Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$. Estimate $F(N)$. In particular, is it true that $ F(N) > N^{1/2-o(1)}? $ | This was studied by Erd\H{o}s, Lev, Rauzy, S\'{a}ndor, and S\'{a}rk"{o}zy \cite{ELRSS99}, where they call such a property 'non-dividing', and prove the explicit bound $ F(N)<3N^{1/2}+1. $ In \cite{Er97b} Erd\H{o}s credits Csaba with a construction that proves $F(N) \gg N^{1/5}$. Such a construction was also given in \c... | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,949 | EP-132 | Erdős Problem #132 | Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$? | Asked by Erd\H{o}s and Pach. Hopf and Pannowitz \cite{HoPa34} proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.
It may be true that there are at least $n^{1-o(1)}$ many such distances. In \cite{Er97e} Erd\H{o}s offers \$100 f... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,950 | EP-137 | Erdős Problem #137 | We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful? | Conjectured by Erd\H{o}s and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erd\H{o}s and Selfridge \cite{ErSe75} proved that the product of $k\geq 3$ consecutive positive integers can never be a perfect power. Erd\H{o}s remarked that this 'seems hopeless at present'.
In \cite{Er82... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,951 | EP-138 | Erdős Problem #138 | Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$. | When $p$ is prime Berlekamp \cite{Be68} has proved $W(p+1)\geq p2^p$. Gowers \cite{Go01} has proved $ W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}. $ The best general lower bound is $W(k)\gg 2^k$, due to Kozik and Shabanov \cite{KoSh16}.
In \cite{Er81} Erd\H{o}s further asks whether $W(k+1)/W(k)\to \infty$, or $W(k+1)-W(k)\to \in... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,952 | EP-141 | Erdős Problem #141 | Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression? | Green and Tao \cite{GrTa08} have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erd\H{o}s called this conjecture 'completely hopeless at present'.
The existence of such progressions for small $k$ has been verified for $k\leq 10$, see the Wikipedia page.... | 3 | open | null | null | 1 | 8 | 0 | 0 | 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": 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"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,953 | EP-142 | Erdős Problem #142 | Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$. | Erd\H{o}s remarked this is 'probably unattackable at present'. In \cite{Er97c} Erd\H{o}s offered \$1000, but given that he elsewhere offered \$5000 just for (essentially) showing that $r_k(N)=o_k(N/\log N)$, that value seems odd. In \cite{Er81} he offers \$10000, stating it is 'probably enormously difficult'.
The best ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,954 | EP-143 | Erdős Problem #143 | Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x
eq y\in A$ and integers $k\geq 1$ we have $ \lvert kx -y\rvert \geq 1. $ Does this imply that $A$ is sparse? In particular, does this imply that $ \sum_{x\in A}\frac{1}{x\log x}<\infty $ or $ \sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)... | Note that if $A$ is a set of integers then the condition implies that $A$ is a primitive set (that is, no element of $A$ is divisible by any other), for which the convergence of $\sum_{n\in A}\frac{1}{n\log n}$ was proved by Erd\H{o}s \cite{Er35}, and the upper bound $ \sum_{n<x}\frac{1}{n}\ll \frac{\log x}{\sqrt{\log\... | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,955 | EP-145 | Erdős Problem #145 | Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$, $ \lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha $ exists? | Erd\H{o}s \cite{Er51} proved this for all $0\leq \alpha \leq 2$, and Hooley \cite{Ho73} extended this to all $\alpha \leq 3$.
Greaves, Harman, and Huxley showed (in Chapter 11 of \cite{GHH97}) that this is true for $\alpha \leq 11/3$. Chan \cite{Ch23c} has extended this to $\alpha \leq 3.75$.
Granville \cite{Gr98} prov... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,956 | EP-146 | Erdős Problem #146 | If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $ \mathrm{ex}(n;H) \ll n^{2-1/r}. $ | Conjectured by Erd\H{o}s and Simonovits \cite{ErSi84}. Open even for $r=2$. Alon, Krivelevich, and Sudakov \cite{AKS03} have proved $ \mathrm{ex}(n;H) \ll n^{2-1/4r}. $ They also prove the full Erd\H{o}s-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one side of the bipartition is $r$.
See a... | 1 | solved | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 146 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/146",
"label": "Erdős Problems #146: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hug... | 2026-09-27T00:00:00 | Erdős Problems #146: disproved (Lean): https://www.erdosproblems.com/146
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,957 | EP-148 | Erdős Problem #148 | Let $F(k)$ be the number of solutions to $ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k}, $ where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$. | The current best bounds known are $ 2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k}, $ where $c>0$ is some absolute constant and $c_0=1.26408\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin \cite{Ko14} and the upper bound to Elsholtz and Planitzer \cite{ElPl21}.
References
[ElPl2... | 1 | open | null | null | 1 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,958 | EP-149 | Erdős Problem #149 | Let $G$ be a graph with maximum degree $\Delta$. Is $G$ the union of at most $\tfrac{5}{4}\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)? | Asked by Erd\H{o}s and Ne\v{s}et\v{r}il in 1985 (see \cite{FGST89}). This is equivalent to asking whether the chromatic number of the square of the line graph $L(G)^2$ is at most $\frac{5}{4}\Delta^2$.
This bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is ... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,959 | EP-151 | Erdős Problem #151 | For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).
Let $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertice... | It is easy to see that $\tau(G) \leq n-\sqrt{n}$. Note also that if $G$ is triangle-free then trivially $\tau(G)\leq n-H(n)$.
This is listed in \cite{Er88} as a problem of Erd\H{o}s and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. There Erd\H{o}s remarked that this conjecture is 'perhaps co... | 1 | open | null | null | 3 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,960 | EP-152 | Erdős Problem #152 | For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1
ot\in A+A$. | There may even be $\gg \lvert A\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature asse... | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | The maintained Erdős Problems source lists problem 152 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/152",
"label": "Erdős Problems #152: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggin... | 2026-09-27T00:00:00 | Erdős Problems #152: proved (Lean): https://www.erdosproblems.com/152
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27T00:00:00",
"kind": "resolution",
"evidence": "maintained_source"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,961 | EP-153 | Erdős Problem #153 | Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $ \frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty $ as $\lvert A\rvert\to \infty$? | A similar problem can be asked for infinite Sidon sets.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The mean-square consecutive-gap conjecture for finite Sidon sumsets rem... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,962 | EP-155 | Erdős Problem #155 | Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have $ F(N+k)\leq F(N)+1 $ for all sufficiently large $N$? | This may even hold with $k\approx \epsilon N^{1/2}$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** For every fixed k, eventual control F(N+k) <= F(N)+1 for the extremal Sido... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,963 | EP-156 | Erdős Problem #156 | Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$? | A question of Erd\H{o}s, S\'{a}rk"{o}zy, and S\'{o}s \cite{ESS94}. It is easy to prove that the greedy construction of a maximal Sidon set in $\{1,\ldots,N\}$ has size $\gg N^{1/3}$. Ruzsa \cite{Ru98b} constructed a maximal Sidon set of size $\ll (N\log N)^{1/3}$.
See also [340].
References
[ESS94] Erd\H{o}s, P. and ... | 1 | open | null | null | 2 | 8 | 0 | 0 | 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": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.