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2,164 | EP-535 | Erdős Problem #535 | Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$. | Erd\H{o}s \cite{Er64} proved that $ f_r(N) \leq N^{\frac{3}{4}+o(1)}, $ and Abbott and Hanson \cite{AbHa70} improved this exponent to $1/2$. Erd\H{o}s \cite{Er64} proved the lower bound $ f_3(N) > N^{\frac{c}{\log\log N}} $ for some constant $c>0$, and conjectured this should also be an upper bound.
Erd\H{o}s writes th... | 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 |
2,165 | EP-536 | Erdős Problem #536 | Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be distinct $a,b,c\in A$ such that $ [a,b]=[b,c]=[a,c], $ where $[a,b]$ denotes the least common multiple? | This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erd\H{o}s \cite{Er62} (with a proof given in \cite{Er70}).
This was also asked by Erd\H{o}s at the 1991 problem session of West Coast Number Theory.
In the comments Weisenberg sketches a construction of a set $A\subseteq... | 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 |
2,166 | EP-538 | Erdős Problem #538 | Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$. Give the best possible upper bound for $ \sum_{n\in A}\frac{1}{n}. $ | Erd\H{o}s observed that $ \sum_{n\in A}\frac{1}{n}\sum_{p\leq N}\frac{1}{p}\leq r\sum_{m\leq N^2}\frac{1}{m}\ll r\log N, $ and hence $ \sum_{n\in A}\frac{1}{n} \ll r\frac{\log N}{\log\log N}. $ See also [536] and [537].",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08... | 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 |
2,167 | EP-539 | Erdős Problem #539 | Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set $ \left\{ \frac{a}{(a,b)}: a,b\in A\right\} $ has size at least $h(n)$. Estimate $h(n)$. | Erd\H{o}s and Szemer\'{e}di proved that $ n^{1/2} \ll h(n) \ll n^{1-c} $ for some constant $c>0$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** ProofCouncil... | 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 |
2,168 | EP-543 | Erdős Problem #543 | Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$ then, with probability $\geq 1/2$, all elements of $G$ can be written as $\sum_{x\in S}x$ for some $S\subseteq A$. Is $ f(N) \leq \log_2 N+o(\log\log N)? $ | Erd\H{o}s and R\'{e}nyi \cite{ErRe65} proved that $ f(N) \leq \log_2N+O(\log\log N). $ Erd\H{o}s believed improving this to $o(\log\log N)$ is impossible.
References
[ErRe65] Erd\H{o}s, P. and R\'{e}nyi, A., Probabilistic methods in group theory. J. Analyse Math. (1965), 127-138.",
"difficulty": "L1"
},{
<!-- LI... | 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",
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"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 543 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/543",
"label": "Erdős Problems #543: 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://huggingfac... | 2026-09-27T00:00:00 | Erdős Problems #543: disproved: https://www.erdosproblems.com/543
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 |
2,169 | EP-544 | Erdős Problem #544 | Show that $ R(3,k+1)-R(3,k)\to\infty $ as $k\to \infty$. Similarly, prove or disprove that $ R(3,k+1)-R(3,k)=o(k). $ | A problem of Erd\H{o}s and S\'{o}s.
This problem is #8 in Ramsey Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Both the divergence o... | 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 |
2,170 | EP-545 | Erdős Problem #545 | Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\binom{n}{2}+t$ edges with $0\leq t<n$ then is $ R(G)\leq R(H), $ where $H$ is the graph formed by connecting a new vertex to $t$ of the vertices of $K_n$? | A question of Erd\H{o}s and Graham. The weaker question of whether $ R(G) \leq 2^{O(m^{1/2})} $ is the subject of [546]. (This is true, and was proved by Sudakov \cite{Su11}.)
LouisD in the comments has noted this fails for small $m$ (in particular for $2\leq m\leq 5$ and $7\leq m\leq 9$).
This problem is #10 in Ramsey... | 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 #545 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/545",
"label": "Erdős Problems #545: open"
},
{
"url": "https://doi.org/10.1017/S1446788700029554",
"label": "E. J. Cockayne and P. J. Lorimer, The Ramsey number for stripes, J. Austral. Math. Soc. 19 (1975), 252-256, doi:10.1017/S1446788700029554."
},
... | 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 |
2,171 | EP-550 | Erdős Problem #550 | Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\ldots,m_k$ then prove that $ R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1. $ | Chv\'{a}tal \cite{Ch77} proved that $R(T,K_m)=(m-1)(n-1)+1$.
This problem is #16 in Ramsey Theory in the graphs problem collection.
References
[Ch77] Chv\'{a}tal, V., Tree-complete graph Ramsey numbers. J. Graph Theory (1977), 93.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (ch... | 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 #550 as “open (Lean)” 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. A Lean formalization of an open statement is not a Lean proof of... | [
{
"url": "https://www.erdosproblems.com/550",
"label": "Erdős Problems #550: open (Lean)"
},
{
"url": "https://arxiv.org/abs/2606.23659",
"label": "Eric Li, A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers, arXiv:2606.23659 (2026)."
},
{
"url": "https... | 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 |
2,172 | EP-552 | Erdős Problem #552 | Determine the Ramsey number $ R(C_4,S_n), $ where $S_n=K_{1,n}$ is the star on $n+1$ vertices.
In particular, is it true that, for any $c>0$, there are infinitely many $n$ such that $ R(C_4,S_n)\leq n+\sqrt{n}-c? $ | A problem of Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp \cite{BEFRS89}. Erd\H{o}s often asked about $R(C_4,S_n)$ in the equivalent formulation of asking for a bound on the minimum degree of a graph which would guarantee the existence of a $C_4$ (see [85]).
It is known that $ n+\sqrt{n}-6n^{11/40} \leq R(C_4,S_n)\l... | 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 |
2,173 | EP-554 | Erdős Problem #554 | Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that $ \lim_{k\to \infty}\frac{R(C_{2n+1};k)}{R(K_3;k)}=0 $ for any $n\geq 2$. | A problem of Erd\H{o}s and Graham. The problem is open even for $n=2$.
This problem is #23 in Ramsey Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**C... | 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 |
2,174 | EP-555 | Erdős Problem #555 | Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of $ R(C_{2n};k). $ | A problem of Erd\H{o}s and Graham. Erd\H{o}s \cite{Er81c} gives the bounds $ k^{1+\frac{1}{2n}}\ll R(C_{2n};k)\ll k^{1+\frac{1}{n-1}}. $ Chung and Graham \cite{ChGr75} showed that $ R(C_4;k)>k^2-k+1 $ when $k-1$ is a prime power and $ R(C_4;k)\leq k^2+k+1 $ for all $k$.
This problem is #24 in Ramsey Theory in the graph... | 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 |
2,175 | EP-557 | Erdős Problem #557 | Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that $ R(T;k)\leq kn+O(1) $ for any tree $T$ on $n$ vertices? | A problem of Erd\H{o}s and Graham. Implied by [548].
This would be best possible since, for example, $R(S_n,k)\geq kn-O(k)$ if $S_n=K_{1,n-1}$ is a star on $n$ vertices.
This problem is #26 in Ramsey Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature re... | 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 557 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/557",
"label": "Erdős Problems #557: proved (Lean)"
},
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"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 #557: proved (Lean): https://www.erdosproblems.com/557
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 |
2,176 | EP-558 | Erdős Problem #558 | Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine $ R(K_{s,t};k) $ where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other. | Chung and Graham \cite{ChGr75} prove the general bounds $ (2\pi\sqrt{st})^{\frac{1}{s+t}}\left(\frac{s+t}{e^2}\right)k^{\frac{st-1}{s+t}}\leq R(K_{s,t};k)\leq (t-1)(k+k^{1/s})^s $ and determined $ R(K_{2,2},k)=(1+o(1))k^2. $ Alon, R\'{o}nyai, and Szab\'{o} \cite{ARS99} have proved that $ R(K_{3,3},k)=(1+o(1))k^3 $ and ... | 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 |
2,177 | EP-560 | Erdős Problem #560 | Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.
Determine $ \hat{R}(K_{n,n}), $ where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each... | We know that $ \frac{1}{60}n^22^n<\hat{R}(K_{n,n})< \frac{3}{2}n^32^n. $ The lower bound (which holds for $n\geq 6$) was proved by Erd\H{o}s and Rousseau \cite{ErRo93}. The upper bound was proved by Erd\H{o}s, Faudree, Rousseau, and Schelp \cite{EFRS78b} and Ne\v{s}et\v{r}il and R"{o}dl \cite{NeRo78}.
Conlon, Fox, and ... | 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 |
2,178 | EP-561 | Erdős Problem #561 | Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.
Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\c... | Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp \cite{BEFRS78} proved this when all the $n_i$ are identical and all the $m_i$ are identical.
This problem is #30 in Ramsey Theory in the graphs problem collection.
References
[BEFRS78] Burr, S. A. and Erd\H{o}s, P. and Faudree, R. J. and Rousseau,
C. C. and Schelp, R. H.... | 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 |
2,179 | EP-562 | Erdős Problem #562 | Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.
Prove that, for $r\geq 3$, $ \log_{r-1} R_r(n) \asy... | A problem of Erd\H{o}s, Hajnal, and Rado \cite{EHR65}. A generalisation of [564].
This problem is #38 in Ramsey Theory in the graphs problem collection.
References
[EHR65] Erd\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.",
"difficul... | 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 |
2,180 | EP-563 | Erdős Problem #563 | Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.
Prove that, for every $0\leq \alpha< 1/2$, $ F(n,\alpha)\sim c_\alpha\log n $... | It is easy to show via the probabilistic method that, for every $0\leq \alpha<1/2$, $ F(n,\alpha)\asymp_\alpha \log n. $ Note that when $\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.
See also... | 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 |
2,181 | EP-564 | Erdős Problem #564 | Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.
Is there some constant $c>0$ such that $ R_3(n) \geq 2^{2^{cn}}? $ | A special case of [562]. A problem of Erd\H{o}s, Hajnal, and Rado \cite{EHR65}, who prove the bounds $ 2^{cn^2}< R_3(n)< 2^{2^{n}} $ for some constant $c>0$.
Erd\H{o}s, Hajnal, M\'{a}t\'{e}, and Rado \cite{EHMR84} have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.
This problem ... | 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 |
2,182 | EP-566 | Erdős Problem #566 | Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\ll m? $ | In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erd\H{o}s, Faudree, Rousseau, and Schelp \cite{EFRS93} have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.
Implies [567].
This problem is #31 in... | 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 |
2,183 | EP-567 | Erdős Problem #567 | Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\ll m? $ | In other words, is $G$ Ramsey size linear? A special case of [566]. In \cite{Er95} Erd\H{o}s specifically asks about the case $G=K_{3,3}$.
The graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\gg n^{3-o(1)}$, see [166].) Bra... | 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 |
2,184 | EP-568 | Erdős Problem #568 | Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, $ R(G,H)\ll m? $ | In other words, is $G$ Ramsey size linear?
This problem is #33 in Ramsey Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No verified t... | 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 |
2,185 | EP-569 | Erdős Problem #569 | Let $k\geq 1$. What is the best possible $c_k$ such that $ R(C_{2k+1},H)\leq c_k m $ for any graph $H$ on $m$ edges without isolated vertices? | This problem is #34 in Ramsey Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The asymptotically sharp eventual coeff... | 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 |
2,186 | EP-571 | Erdős Problem #571 | Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that $ \mathrm{ex}(n;G)\asymp n^{\alpha}. $ | A problem of Erd\H{o}s and Simonovits.
Bukh and Conlon \cite{BuCo18} proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.
A rational $\alpha\in [1,2)$ for which this holds is known as a Tur\'{a}n exponent. Known Tur\'{a}n ex... | 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 571 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/571",
"label": "Erdős Problems #571: 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 #571: proved (Lean): https://www.erdosproblems.com/571
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 |
2,187 | EP-572 | Erdős Problem #572 | Show that for $k\geq 3$ $ \mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}. $ | It is easy to see that $\mathrm{ex}(n;C_{2k+1})=\lfloor n^2/4\rfloor$ for any $k\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erd\H{o}s and Klein \cite{Er38} proved $\mathrm{ex}(n;C_4)\asymp n^{3/2}$.
Erd\H{o}s \cite{Er64c} and Bondy and Simonovits \cite{BoSi74} showed that $ \mathrm{ex}(n;C_... | 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 |
2,188 | EP-573 | Erdős Problem #573 | Is it true that $ \mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}? $ | A problem of Erd\H{o}s and Simonovits, who proved that $ \mathrm{ex}(n;\{C_4,C_5\})=(n/2)^{3/2}+O(n). $ K"{o}v\'{a}ri, S\'{o}s, and Tur\'{a}n \cite{KST54} proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\sim (n/2)^{3/2}$. This problem is therefore asking whether th... | 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 |
2,189 | EP-574 | Erdős Problem #574 | Is it true that, for $k\geq 2$, $ \mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}. $ | A problem of Erd\H{o}s and Simonovits.
See also [573] for the specific case of $k=2$.
This problem is #49 in Extremal Graph Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-... | 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 574 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/574",
"label": "Erdős Problems #574: 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://huggingfac... | 2026-09-27T00:00:00 | Erdős Problems #574: disproved: https://www.erdosproblems.com/574
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 |
2,190 | EP-575 | Erdős Problem #575 | If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$.
Is it true ... | A problem of Erd\H{o}s and Simonovits.
See also [180].
This problem is #51 in Extremal Graph Theory in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literatur... | 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 575 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/575",
"label": "Erdős Problems #575: 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://huggingfac... | 2026-09-27T00:00:00 | Erdős Problems #575: disproved: https://www.erdosproblems.com/575
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 |
2,191 | EP-576 | Erdős Problem #576 | Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of $ \mathrm{ex}(n;Q_k). $ | Erd\H{o}s and Simonovits \cite{ErSi70} proved that $ (\tfrac{1}{2}+o(1))n^{3/2}\leq \mathrm{ex}(n;Q_3) \ll n^{8/5}. $ (In \cite{ErSi70} they mention that Erd\H{o}s had originally conjectured that $ \mathrm{ex}(n;Q_3)\gg n^{5/3}$.) Erd\H{o}s and Simonovits also proved that, if $G$ is the graph $Q_3$ with a missing edge,... | 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 |
2,192 | EP-579 | Erdős Problem #579 | Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\delta n^2$ edges then $G$ contains an independent set of size $\gg_\delta n$. | A problem of Erd\H{o}s, Hajnal, S\'{o}s, and Szemer\'{e}di, who could prove this is true for $\delta>1/8$.
See also [533] and the entry in the graphs problem collection.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classificat... | 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 |
2,193 | EP-584 | Erdős Problem #584 | Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that
{UL}
{LI}$H_1$ has $\gg \delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and
{LI... | A problem of Erd\H{o}s, Duke, and R"{o}dl. Duke and Erd\H{o}s \cite{DuEr83}, who proved the first if $n$ is sufficiently large depending on $\delta$. The real challenge is to prove this when $\delta=n^{-c}$ for some $c>0$. Duke, Erd\H{o}s, and R"{o}dl \cite{DER84} proved the first statement with a $\delta^5$ in place o... | 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 |
2,194 | EP-585 | Erdős Problem #585 | What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set? | Pyber, R"{o}dl, and Szemer\'{e}di \cite{PRS95} constructed such a graph with $\gg n\log\log n$ edges.
Chakraborti, Janzer, Methuku, and Montgomery \cite{CJMM24} have shown that such a graph can have at most $n(\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\geq 2$ ... | 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 |
2,195 | EP-588 | Erdős Problem #588 | Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that $ f_k(n)=o(n^2) $ for $k\geq 4$? | A generalisation of [101] (which asks about $k=4$).
The restriction to $k\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Gr"{u}nbaum, and Sloane \cite{BGS74} and F"{u}redi and Pal\'{a}sti \cite{FuPa84} for constructions which show that $f_3(n)\geq(1/6+o(1))n^2$.)
For $k\geq 4$,... | 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 |
2,196 | EP-589 | Erdős Problem #589 | Let $g(n)$ be maximal such that in any set of $n$ points in $\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$. | The trivial greedy algorithm gives $g(n)\gg n^{1/2}$. A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\leq l<k$.
Erd\H{o}s thought that $g(n) \gg n$, but in fact $g(n)=o(n)$, which follows from the density Hales-Jewett theorem proved... | 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 |
2,197 | EP-591 | Erdős Problem #591 | Let $\alpha$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$? | For comparison, Specker \cite{Sp57} proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$. Chang proved this property holds when $\alpha=\omega^\omega$ (see [590]).
This is true and was proved independently by Schipperus \cite{Sc10} and Darby.
See also [118], and [592] ... | 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 591 as “proved” 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/591",
"label": "Erdős Problems #591: proved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #591: proved: https://www.erdosproblems.com/591
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 |
2,198 | EP-592 | Erdős Problem #592 | Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$. | Such $\alpha$ are called partition ordinals.
{UL}
{LI}Specker \cite{Sp57} proved this holds for $\beta=2$ and not for $3\leq \beta <\omega$.{/LI}
{LI} Chang \cite{Ch72} proved this holds for $\beta=\omega$.{/LI}
{LI} Galvin and Larson \cite{GaLa74} have shown that if $\beta\geq 3$ has this property then $\beta$ must be... | 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 |
2,199 | EP-593 | Erdős Problem #593 | Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$. | Similar problems were investigated by Erd\H{o}s, Galvin, and Hajnal \cite{EGH75}. Erd\H{o}s claims that for graphs the problem is completely solved: a graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.
References
[EGH75] Erd\H{o}s, P. and Galvi... | 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 |
2,200 | EP-595 | Erdős Problem #595 | Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs? | A problem of Erd\H{o}s and Hajnal. Folkman \cite{Fo70} and Ne\v{s}et\v{r}il and R"{o}dl \cite{NeRo75} have proved that for every $n\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.
See also [582] and [596].
References
[Fo70] Folkman, Jon, Graphs with monochromatic ... | 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 |
2,201 | EP-596 | Erdős Problem #596 | For which graphs $G_1,G_2$ is it true that
{UL}
{LI} for every $n\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet{/LI}
{LI} for every graph $H$ without a $G_1$ there is an $\aleph_0$-colouring of the edges of $H$ without a monochr... | Erd\H{o}s and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Ne\v{s}et\v{r}il and R"{o}dl established the first property and Erd\H{o}s and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees).
Whether 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 |
2,202 | EP-597 | Erdős Problem #597 | Let $G$ be a graph on at most $\aleph_1$ vertices which contains no $K_4$ and no $K_{\aleph_0,\aleph_0}$ (the complete bipartite graph with $\aleph_0$ vertices in each class). Is it true that $ \omega_1^2 \to (\omega_1\omega, G)^2? $ What about finite $G$? | Erd\H{o}s and Hajnal proved that $\omega_1^2 \to (\omega_1\omega,3)^2$. Erd\H{o}s originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\omega_1^2
ot\to (\omega_1\omega, K_{\aleph_0,\aleph_0})^2$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 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 |
2,203 | EP-598 | Erdős Problem #598 | Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X\subseteq m$ with $\lvert X\rvert=\kappa$ contains subsets of all possible colours?
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The ZFC status is open; unverified forum reports describe a positive special case and large-cardinal relative consistency statements.
**Ver... | 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 |
2,204 | EP-600 | Erdős Problem #600 | Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\geq 2$. Is it true that $ e(n,r+1)-e(n,r)\to \infty $ as $n\to \infty$? Is it true that $ \frac{e(n,r+1)}{e(n,r)}\to 1 $... | Ruzsa and Szemer\'{e}di \cite{RuSz78} proved that $e(n,r)=o(n^2)$ for any fixed $r$.
See also [80].
References
[RuSz78] Ruzsa, I. Z. and Szemer\'{e}di, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq.,
Keszthely, 1976), Vol. II (1978), 939-945.",
"diffic... | 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 |
2,205 | EP-601 | Erdős Problem #601 | For which limit ordinals $\alpha$ is it true that if $G$ is a graph with vertex set $\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\alpha$? | A problem of Erd\H{o}s, Hajnal, and Milner \cite{EHM70}, who proved this is true for $\alpha < \omega_1^{\omega+2}$.
In \cite{Er82e} Erd\H{o}s offers \$250 for showing what happens when $\alpha=\omega_1^{\omega+2}$ and \$500 for settling the general case.
Larson \cite{La90} proved this is true for all $\alpha<2^{\aleph... | 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 |
2,206 | EP-602 | Erdős Problem #602 | Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i
eq j$ we have $\lvert A_i\cap A_j\rvert$ finite and $
eq 1$. Is there a $2$-colouring of $\cup A_i$ such that no $A_i$ is monochromatic? | A problem of Komj\'{a}th. The existence of such a $2$-colouring is sometimes known as Property B.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** Property B i... | 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 |
2,207 | EP-603 | Erdős Problem #603 | Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert
eq 2$ for all $i
eq j$. Find the smallest cardinal $C$ such that $\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic. | A problem of Komj\'{a}th. If instead we have $\lvert A_i\cap A_j\rvert
eq 1$ then Komj\'{a}th showed that this is possible with at most $\aleph_0$ colours.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATU... | 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 603 as “solved” 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/603",
"label": "Erdős Problems #603: solved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #603: solved: https://www.erdosproblems.com/603
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 |
2,208 | EP-604 | Erdős Problem #604 | Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that $ \#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}? $ Or even $\gg n/\sqrt{\log n}$? | The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.
It may be true that there are $\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \cite{Er75f} ... | 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 |
2,209 | EP-609 | Erdős Problem #609 | Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$. | A problem of Erd\H{o}s and Graham. The edges of $K_{2^n}$ can be $n$-coloured to avoid odd cycles of any length. It can be shown that $C_5$ and $C_7$ can be avoided for large $n$.
Chung \cite{Ch97} asked whether $f(n)\to \infty$ as $n\to \infty$. Day and Johnson \cite{DaJo17} proved this is true, and that $ f(n)\geq 2^... | 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 |
2,210 | EP-610 | Erdős Problem #610 | For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).
Estimate $\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then $ \tau(G) \leq n-\omega(n)\sqrt{n} $ for some $\omega(n)\to \inf... | A problem of Erd\H{o}s, Gallai, and Tuza \cite{EGT92}, who proved that $ \tau(G) \leq n-\sqrt{2n}+O(1). $ This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\sqrt{n\log n})$, which follows from the lower bound for $R(3,k)$ by Kim \cite{Ki95} (see [165]).
Indeed, Erd... | 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 610 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/610",
"label": "Erdős Problems #610: 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 #610: proved (Lean): https://www.erdosproblems.com/610
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 |
2,211 | EP-611 | Erdős Problem #611 | For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).
Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\tau(G)=o_c(n)$?
Similarly, estimate for $c>0$ the minimal $k_c(... | A problem of Erd\H{o}s, Gallai, and Tuza \cite{EGT92}, who proved for the latter question that $k_c(n) \geq n^{c'/\log\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\tau(G) \leq n-(kn)^{1/2}$. Bollob\'{a}s and Erd\H{o}s proved that if every maximal clique has at least $n+3-2\sqrt{n}$ vertic... | 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 |
2,212 | EP-612 | Erdős Problem #612 | Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\mid d$ then $ D\leq \frac{2(r-1)(3r+2)}{2r^2-1}\frac{n}{d}+O(1), $ and if $G$ contains no $K_{2r+1}$ and $3r-1 \mid d$ then $ D\leq \frac{3r-1}{r}\frac{n}{d}+O(1). $ | A problem of Erd\H{o}s, Pach, Pollack, and Tuza \cite{EPPT89}, who gave constructions showing that the above bounds would be sharp, and proved the case $2r+1=3$. It is known (see \cite{EPPT89} for example) that any connected graph on $n$ vertices with minimum degree $d$ has diameter $ D\leq 3\frac{n}{d+1}+O(1). $ This ... | 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 #612 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/612",
"label": "Erdős Problems #612: open"
},
{
"url": "https://doi.org/10.1016/j.jctb.2020.06.003",
"label": "É. Czabarka, I. Singgih and L. A. Székely, Counterexamples to a conjecture of Erdős, Pach, Pollack and Tuza, J. Combin. Theory Ser. B (2021), 38-... | 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 |
2,213 | EP-614 | Erdős Problem #614 | Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The general extremal function remains open, with a finite-forbidden-family Turán reformulation for each fixed k.
**Verified partial progres... | 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 |
2,214 | EP-616 | Erdős Problem #616 | Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.
Determine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at m... | Erd\H{o}s, Hajnal, and Tuza \cite{EHT91} proved that this $t$ satisfies $ \frac{3}{16}r+\frac{7}{8}\leq t \leq \frac{1}{5}r. $
References
[EHT91] Erd\H{o}s, Paul and Hajnal, Andr\'{a}s and Tuza, Zsolt, Local constraints ensuring small representing sets. J. Combin. Theory Ser. A (1991), 78-84.",
"difficulty": "L1... | 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 |
2,215 | EP-619 | Erdős Problem #619 | For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).
Is it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$? | A problem of Erd\H{o}s, Gy\'{a}rf\'{a}s, and Ruszink\'{o} \cite{EGR98} who proved that $h_3(G)\leq n$ and $h_5(G) \leq \frac{n-1}{2}$ and there exist connected graphs $G$ on $n$ vertices with $h_3(G)\geq n-c$ for some constant $c>0$.
If we omit the condition that the graph must remain triangle-free then Alon, Gy\'{a}rf... | 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 619 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/619",
"label": "Erdős Problems #619: 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 #619: solved (Lean): https://www.erdosproblems.com/619
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 |
2,216 | EP-620 | Erdős Problem #620 | If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain? | This was first asked by Erd\H{o}s and Rogers \cite{ErRo62}, and is generally known as the Erd\H{o}s-Rogers problem. Let $f(n)$ be such that every such graph contains a triangle-free subgraph with at least $f(n)$ vertices.
It is now known that $f(n)=n^{1/2+o(1)}$. Bollob\'{a}s and Hind \cite{BoHi91} proved $ n^{1/2} \ll... | 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 |
2,217 | EP-623 | Erdős Problem #623 | Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)
ot\in A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)
ot\in Y$? | A problem of Erd\H{o}s and Hajnal \cite{ErHa58}, who proved that if $\lvert X\rvert <\aleph_\omega$ then the answer is no. Erd\H{o}s suggests in \cite{Er99} that this problem is 'perhaps undecidable'.
References
[Er99] Erd\H{o}s, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (199... | 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 |
2,218 | EP-624 | Erdős Problem #624 | Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have $ \{ f(A) : A\subseteq Y\}=X. $ Prove that $ H(n)-\log_2 n \to \infty. $ | A problem of Erd\H{o}s and Hajnal \cite{ErHa68} who proved that $ \log_2 n \leq H(n) < \log_2n +(3+o(1))\log_2\log_2n. $ Erd\H{o}s said that even the weaker statement that for $n=2^k$ we have $H(n)\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\frac{n-1}{2}$ ... | 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 |
2,219 | EP-625 | Erdős Problem #625 | The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\chi(G)$ denote the chromatic number.
If $G$ is a random graph with $n$ vertices and each edge included independent... | A problem of Erd\H{o}s and Gimbel (see also \cite{Gi16}). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erd\H{o}s offered \$100 for a proof that this is true, and \$1000 for a proof that this is false (although later told Gimbel that \$1000 was perhaps too much).
It is known that almost surel... | 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 625 as “solved” 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/625",
"label": "Erdős Problems #625: solved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #625: solved: https://www.erdosproblems.com/625
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 |
2,220 | EP-626 | Erdős Problem #626 | Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\leq m$). Does $ \lim_{n\to \infty}\frac{g_k(n)}{\log n} $ exist?
Conversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertice... | It is known that $ \frac{1}{4\log k}\log n\leq g_k(n) \leq \frac{2}{\log(k-2)}\log n+1, $ the lower bound due to Kostochka \cite{Ko88} and the upper bound to Erd\H{o}s \cite{Er59b}.
Erd\H{o}s \cite{Er59b} proved that $ \lim_{n\to \infty}\frac{\log h^{(m)}(n)}{\log n}\gg \frac{1}{m} $ and, for odd $m$, $ \lim_{n\to \inf... | 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 |
2,221 | EP-627 | Erdős Problem #627 | Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\chi(G)/\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does $ \lim_{n\to\infty}\frac{f(n)}{n/(\log n)^2} $ exist? | Tutte and Zykov \cite{Zy52} independently proved that for every $k$ there is a graph with $\omega(G)=2$ and $\chi(G)=k$. Erd\H{o}s \cite{Er61d} proved that for every $n$ there is a graph on $n$ vertices with $\omega(G)=2$ and $\chi(G)\gg n^{1/2}/\log n$, whence $f(n) \gg n^{1/2}/\log n$.
Erd\H{o}s \cite{Er67c} proved 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 |
2,222 | EP-629 | Erdős Problem #629 | The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.
Deter... | A problem of Erd\H{o}s, Rubin, and Taylor \cite{ERT80}, who proved that $ 2^{k-1}<n(k) <k^22^{k+2}. $ They also prove that if $m(k)$ is the size of the smallest family of $k$-sets without property B (i.e. the smallest number of $k$-sets in a graph with chromatic number $3$) then $m(k)\leq n(k)\leq m(k+1)$. Bounds on $m... | 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 |
2,223 | EP-633 | Erdős Problem #633 | Classify those triangles which can only be cut into a square number of congruent triangles. | Erd\H{o}s' question was reported by Soifer \cite{So09c}. It is easy to see (see for example \cite{So09}) that any triangle can be cut into $n^2$ congruent triangles (for any $n\geq 1$). Soifer \cite{So09b} proved that there exists at least one triangle (e.g. one with sides $\sqrt{2},\sqrt{3},\sqrt{4}$) which can only b... | 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 633 as “solved” 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/633",
"label": "Erdős Problems #633: solved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #633: solved: https://www.erdosproblems.com/633
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 |
2,224 | EP-634 | Erdős Problem #634 | Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles. | Erd\H{o}s' question was reported by Soifer \cite{So09c}. It is easy to see that all square numbers have this property (in fact for square numbers any triangle will do). Soifer \cite{So09c} has shown that numbers of the form $2n^2,3n^2,6n^2,n^2+m^2$ also have this property. Beeson has shown (see the slides below) that $... | 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 |
2,225 | EP-635 | Erdős Problem #635 | Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a
mid b$. How large can $\lvert A\rvert$ be? Is it true that $ \lvert A\rvert \leq \left(\frac{1}{2}+o_t(1)\right)N? $ | Asked by Erd\H{o}s in a letter to Ruzsa in around 1980. Erd\H{o}s observes that when $t=1$ the maximum possible is $ \lvert A\rvert=\left\lfloor\frac{N+1}{2}\right\rfloor, $ achieved by taking $A$ to be all odd numbers in $\{1,\ldots,N\}$. He also observes that when $t=2$ there exists such an $A$ with $ \lvert A\rvert ... | 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 |
2,226 | EP-638 | Erdős Problem #638 | Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.
Is it true that for every infinite cardinal $\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges ... | Erd\H{o}s writes 'if the answer is affirmative many extensions and generalisations will be possible'.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The maintained Erdős Proble... | 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 #638 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/638",
"label": "Erdős Problems #638: open"
},
{
"url": "https://www.starfleetmath.com/",
"label": "Star Fleet Math, public formal proof bundle for Erdős Problem #638, checked 2026-08-17."
},
{
"url": "https://www.erdosproblems.com/638",
"label"... | 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 |
2,227 | EP-640 | Erdős Problem #640 | Is there some function $f$ such that for all $k\geq 3$ if a finite graph $G$ has chromatic number $\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\geq k$?
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The high-chromatic odd-cycle-span question remains open for k>3.
**Verified partial progress.**
- The k=3 case is immediate from non-bipartiteness.
- Stein... | 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 |
2,228 | EP-642 | Erdős Problem #642 | Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\ll n$? | A problem of Hamburger and Szegedy.
Chen, Erd\H{o}s, and Staton \cite{CES96} proved $f(n) \ll n^{3/2}$. Dragani\'{c}, Methuku, Munh\'{a} Correia, and Sudakov \cite{DMMS24} have improved this to $ f(n) \ll n(\log n)^8. $
References
[CES96] Chen, Guantao and Erd\H{o}s, Paul and Staton, William, Proof of a conjecture o... | 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 |
2,229 | EP-643 | Erdős Problem #643 | Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that $ A\cup B= C\cup D $ and $ A\cap B=C\cap D=\emptyset. $ Estimate $f(n;t)$ - in particular, is it true that for $t\geq 3$ $ f(n;t)=(1+o(1))\binom{n}{t-1}? $ | For $t=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1/2+o(1))n^{3/2}$.
F"{u}redi \cite{Fu84} proved that $f(n;3) \ll n^2$ and $f(n;3) > \binom{n}{2}$ for infinitely many $n$. Pikhurko and Verstra"{e}te \cite{PiVe09} have proved $f(n;3)\leq \frac{13}{9}\binom{n}{2... | 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 |
2,230 | EP-644 | Erdős Problem #644 | Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\{x,y\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that $ f(k,7)=(1+o(1)... | A problem of Erd\H{o}s, Fon-Der-Flaass, Kostochka, and Tuza \cite{EFKT92}, who proved that $f(k,3)=2k$ and $f(k,4)=\lfloor 3k/2\rfloor$ and $f(k,5)=\lfloor 5k/4\rfloor$, and further that $f(k,6)=k$.
References
[EFKT92] Erd"{o}s, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and
Tuza, Zs., Small transversals in unifo... | 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 |
2,231 | EP-650 | Erdős Problem #650 | Let $f(m)$ be such that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert=m$ then every interval in $[1,\infty)$ of length $2N$ contains $\geq f(m)$ many distinct integers $b_1,\ldots,b_r$ where each $b_i$ is divisible by some $a_i\in A$, where $a_1,\ldots,a_r$ are distinct.
Estimate $f(m)$. In particular is it true t... | Erd\H{o}s and Sar\'{a}nyi \cite{ErSa59} proved that $f(m)\gg m^{1/2}$.
References
[ErSa59] Erd\H{o}s, P. and Sar\'{a}nyi, Megjegyz\'{e}sek egy versenyfeladathoz. Matematikai Lapok (1959).",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Cla... | 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 650 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/650",
"label": "Erdős Problems #650: 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 #650: solved (Lean): https://www.erdosproblems.com/650
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 |
2,232 | EP-652 | Erdős Problem #652 | Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j
eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots \leq R(x_n). $ Let $\alpha_k$ be minimal such that, for all large enough $n$, there exists a set of $n$ points with $R(x_k)<\alpha_kn^{1/2}$. Is it true that $\alpha_k\t... | It is trivial that $R(x_1)=1$ is possible, and that $R(x_2) \ll n^{1/2}$ is also possible, but we always have $ R(x_1)R(x_2)\gg n. $ Erd\H{o}s originally conjectured that $R(x_3)/n^{1/2}\to \infty$ as $n\to \infty$, but Elekes proved that for every $k$ and $n$ sufficiently large there exists some set of $n$ points with... | 1 | solved | null | null | 6 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 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 652 as “proved” 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/652",
"label": "Erdős Problems #652: proved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #652: proved: https://www.erdosproblems.com/652
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 |
2,233 | EP-653 | Erdős Problem #653 | Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j
eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots \leq R(x_n). $ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$? | Erd\H{o}s and Fishburn proved $g(n)>\frac{3}{8}n$ and Csizmadia proved $g(n)>\frac{7}{10}n$. Both groups proved $g(n) < n-cn^{2/3}$ for some constant $c>0$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Cur... | 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 |
2,234 | EP-654 | Erdős Problem #654 | Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$? | It is clear that every point has at least $\frac{n-1}{3}$ distinct distances to other points in the set.
In \cite{Er87b} and \cite{ErPa90} Erd\H{o}s and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker ... | 1 | open | 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 still lists #654 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/654",
"label": "Erdős Problems #654: open"
},
{
"url": "https://arxiv.org/abs/2602.10177",
"label": "T. Feng et al., Towards Autonomous Mathematics Research, arXiv:2602.10177 (2026), with associated Aletheia Erdős solutions."
},
{
"url": "https://g... | 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 |
2,235 | EP-655 | Erdős Problem #655 | Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\frac{n}{2} $ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large? | A problem of Erd\H{o}s and Pach. It is easy to see that this assumption implies that there are at least $\frac{n-1}{2}$ distinct distances determined by every point.
Zach Hunter has observed that taking $n$ points equally spaced on a circle disproves this conjecture. In the spirit of related conjectures of Erd\H{o}s an... | 1 | partially_solved | 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 |
2,236 | EP-657 | Erdős Problem #657 | Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\to \infty$? | In \cite{Er73} Erd\H{o}s attributes this problem (more generally in $\mathbb{R}^k$) to himself and Davies. In \cite{Er97e} he does not mention Davis, but says this problem was investigated by himself, F"{u}redi, Ruzsa, and Pach.
In \cite{Er73} Erd\H{o}s says it is not even known in $\mathbb{R}$ whether $f(n)\to \infty$... | 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 |
2,237 | EP-660 | Erdős Problem #660 | Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $ (1-o(1))\frac{n}{2} $ many distinct distances between the $x_i$? | For the similar problem in $\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman \cite{Al63} (see [93]). In \cite{Er75f} Erd\H{o}s claims that Altman proved that the vertices determine $\gg n$ many distinct distances, but gives no reference.
References
[Al63] Altman, E., On a problem of P. Erd... | 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 |
2,238 | EP-661 | Erdős Problem #661 | Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $ o\left(\frac{n}{\sqrt{\log n}}\right)? $ | One can also ask this for points in $\mathbb{R}^3$. In $\mathbb{R}^4$ Lenz observed that there are $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.
More generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minim... | 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 |
2,239 | EP-662 | Erdős Problem #662 | Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For example $f(1)=6$, $f(\sqrt{3})=12$, and $f(3)=18$.
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that $d(x_i,x_j)\geq 1$ for all $i
eq j$. Is it true that, provided $n$ is ... | A problem of Erd\H{o}s, Lov\'{a}sz, and Vesztergombi.
This is essentially verbatim the problem description in \cite{Er97e}, but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.
Erd\H{o}s also goes on to write 'Perhaps the following stronger c... | 1 | partially_solved | 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 |
2,240 | EP-663 | Erdős Problem #663 | Let $k\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is sufficiently large, we have $ q(n,k)<(1+o(1))\log n? $ | A problem of Erd\H{o}s and Pomerance.
The bound $q(n,k)<(1+o(1))k\log n$ is easy. It may be true this improved bound holds even up to $k=o(\log n)$.
A heuristic argument in favour of this is provided by Tao in the comments.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 202... | 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 |
2,241 | EP-665 | Erdős Problem #665 | A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert <n$ and every pair of distinct elements $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$.
Is there a constant $C>0$ and, for all large $n$, a pairwise balanced desi... | A problem of Erd\H{o}s and Larson \cite{ErLa82}. In general, as Erd\H{o}s asks in \cite{Er97f}, find the slowest growing function $h$ such that, for all large $n$, there exists a pairwise balanced design with $ \lvert A_i\rvert > n^{1/2}-h(n) $ for all $1\leq i\leq m$.
The problem above asks whether $h(n)\ll 1$. Erd\H{... | 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 |
2,242 | EP-667 | Erdős Problem #667 | Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.
Is $ c(p,q)=\liminf \frac{\log H(n)}{\log n} $ a strictly increasing function of $q$ for $1\leq q... | A problem of Erd\H{o}s, Faudree, Rousseau, and Schelp.
When $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example $ \frac{1}{p-1}\leq c(p,1) \leq \frac{2}{p+1}. $ It is easy to see that if $q=\binom{p-1}{2}+1$ then $c(p,q)=1$. Erd\H{o}s, Faudree, Rousseau, and Schelp have shown that $c(p... | 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 |
2,243 | EP-668 | Erdős Problem #668 | Is it true that the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\to\infty$? Is it always $>1$ for $n>3$? | In fact this is $=1$ also for $n=4$, the unique example given by two equilateral triangles joined by an edge.
Computational evidence of Engel, Hammond-Lee, Su, Varga, and Zs\'{a}mboki \cite{EHSVZ25} and Alexeev, Mixon, and Parshall \cite{AMP25} suggests that this count is $=1$ for various other $5\leq n\leq 21$ (althou... | 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 |
2,244 | EP-669 | Erdős Problem #669 | Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.
Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\... | Trivially $f_k(n)\leq F_k(n)$ and $f_2(n)=F_2(n)=\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Gr"{u}nbaum, and Sloane \cite{BGS74} have proved that $ f_3(n)=\frac{n^2}{6}-O(n) $ and $ F_3(n)=\frac{n^2}{6}-O(n). $ There is a trivial upper bound of $F_k(n) \leq \binom{n}{2}... | 1 | partially_solved | 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 |
2,245 | EP-670 | Erdős Problem #670 | Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$? | The lower bound of $\binom{n}{2}$ for the diameter is trivial. Erd\H{o}s \cite{Er97f} proved the claim when $d=1$.
References
[Er97f] Erd\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Liter... | 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 |
2,246 | EP-671 | Erdős Problem #671 | Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\leq i'\leq n$ with $i
eq i'$. We similarly define $ \mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i^n)p_i^n(x), $ the unique polynomial of de... | Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\to \mathbb{R}$ there exists some $x\in [-1,1]$ where $ \limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty $ and yet $ \mathcal{L}^nf(x) \to f(x)? $ Is there such a sequence such that $ \limsup_{n\to \infty} \sum_{1\leq i\... | 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 |
2,247 | EP-675 | Erdős Problem #675 | We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq a\leq n$, $ a\in A\quad\textrm{ if and only if }\quad a+t_n\in A. $ {UL} | {LI}Does the set of the sums of two squares have the translation property?{/LI}
{LI}If we partition all primes into $P\sqcup Q$, such that each set contains $\gg x/\log x$ many primes $\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?{/LI}
{LI}If $A... | 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 |
2,248 | EP-676 | Erdős Problem #676 | Is every sufficiently large integer of the form $ ap^2+b $ for some prime $p$ and integer $a\geq 1$ and $0\leq b<p$? | The sieve of Eratosthenes implies that almost all integers are of this form, and the Brun-Selberg sieve implies the number of exceptions in $[1,x]$ is $\ll x/(\log x)^c$ for some constant $c>0$. Erd\H{o}s \cite{Er79} believed it is 'rather unlikely' that all large integers are of this form.
What if the condition that $... | 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 |
2,249 | EP-677 | Erdős Problem #677 | Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.
Is it true that for all $m\geq n+k$ $ M(n,k)
eq M(m,k)? $ | The Thue-Siegel theorem implies that, for fixed $k$, there are only finitely many $m,n$ such that $m\geq n+k$ and $M(n,k)=M(m,k)$.
In general, how many solutions does $M(n,k)=M(m,l)$ have when $m\geq n+k$ and $l>1$? Erd\H{o}s expects very few (and none when $l\geq k$).
The only solutions Erd\H{o}s knew were $M(4,3)=M(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 |
2,250 | EP-679 | Erdős Problem #679 | Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k} $ for all $k<n$ which are sufficiently large depending on $\epsilon$ only?
Can one show the stronger version with $ \omega(n-k) < \f... | One can ask similar questions for $\Omega$, the number of prime factors with multiplicity, where $\log k/\log\log k$ is replaced by $\log_2k$.
See also [248] and [413].
In the comments DottedCalculator has disproved the second stronger version, proving that in fact for all large $n$ there exists $k<n$ such that $ \omeg... | 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 |
2,251 | EP-680 | Erdős Problem #680 | Is it true that, for all sufficiently large $n$, there exists some $k$ such that $ p(n+k)>k^2+1, $ where $p(m)$ denotes the least prime factor of $m$?
Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant? | This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.
Erd\H{o}s observed that Cramer's conjecture $ \limsup_{k\to \infty} \frac{p_{k+1}-p_k}{(\log k)^2}=1 $ implies that for all $\epsilon>0... | 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 |
2,252 | EP-681 | Erdős Problem #681 | Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$? | Related to questions of Erd\H{o}s, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The requ... | 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 |
2,253 | EP-683 | Erdős Problem #683 | Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$, satisfies $ P\left(\binom{n}{k}\right)\geq \min(n-k+1, k^{1+c}) $ for some constant $c>0$? | A theorem of Sylvester and Schur (see \cite{Er34}) states that $P(\binom{n}{k})>k$ if $k\leq n/2$. Erd\H{o}s \cite{Er55d} proved that there exists some $c>0$ such that, whenever $k\leq n/2$, $ P\left(\binom{n}{k}\right)\gg k\log k. $ Erd\H{o}s \cite{Er79d} writes it 'seems certain' that this holds for every $c>0$, with... | 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 |
2,254 | EP-684 | Erdős Problem #684 | For $0\leq k\leq n$ write $ \binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.
Let $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$. | A classical theorem of Mahler states that for any $\epsilon>0$ and integers $k$ and $l$ then, writing $ (n+1)\cdots (n+k) = ab $ where the only primes dividing $a$ are $\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\epsilon}$ for all sufficiently large (depending on $\epsilon,k,l$) $n$.
Mahler's ... | 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 |
2,255 | EP-685 | Erdős Problem #685 | Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is $ (1+o(1))k\sum_{k<p<n}\frac{1}{p}? $ Or perhaps even when $k \geq (\log n)^c$? | It is trivial that the number of prime factors is $ >\frac{\log \binom{n}{k}}{\log n}, $ and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTI... | 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 |
2,256 | EP-686 | Erdős Problem #686 | Can every integer $N\geq 2$ be written as $ N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)} $ for some $k\geq 2$ and $m\geq n+k$? | If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?
See also [677].",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** T... | 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 |
2,257 | EP-687 | Erdős Problem #687 | Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\pmod{p}$.
Give good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\ll x^{1+o(1)}$? | This function (associated with Jacobsthal) is closely related to the problem of gaps between primes (see [4]). The best known upper bound is due to Iwaniec \cite{Iw78}, $ Y(x) \ll x^2. $ The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao \cite{FGKMT18}, $ Y(x) \gg x\frac{\log x\log\log\log x}{\log\l... | 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 |
2,258 | EP-688 | Erdős Problem #688 | Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}<p\leq n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p\pmod{p}$.
Estimate $\epsilon_n$ - in particular is it true that $\epsilon_n=o(1)$? | Erd\H{o}s could prove $ \epsilon_n \gg \frac{\log\log\log n}{\log\log n}. $ See also [687] and [689].",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** Whether ... | 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 |
2,259 | EP-689 | Erdős Problem #689 | Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$ satisfies at least two of the congruences $\equiv a_p\pmod{p}$? | One can ask a similar question replacing $2$ by any fixed integer $r$ (provided $n$ is sufficiently large depending on $r$).
See also [687] and [688].
This problem (with $2$ replaced by $10$) is Problem 45 on Green's open problems list.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review... | 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 |
2,260 | EP-690 | Erdős Problem #690 | Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$).
For fixed $k\geq 1$ is $d_k(p)$ unimodular in $p$? That is, it first increases in $p$ until its maximum then decreases. | Erd\H{o}s believes that this is not possible, but could not disprove it. He could show that $p_k$ is about $e^{e^k}$ for almost all $n$, but the maximal value of $d_k(p)$ is assumed for much smaller values of $p$, at $ p=e^{(1+o(1))k}. $ A similar question can be asked if we consider the density of integers whose $k$th... | 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 690 as “solved” 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/690",
"label": "Erdős Problems #690: solved"
},
{
"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.c... | 2026-09-27T00:00:00 | Erdős Problems #690: solved: https://www.erdosproblems.com/690
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 |
2,261 | EP-691 | Erdős Problem #691 | Given $A\subseteq \mathbb{N}$ let $M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and sufficient condition on $A$ for $M_A$ to have density $1$. | A sequence $A$ for which $M_A$ has density $1$ is called a Behrend sequence.
If $A$ is a set of prime numbers (or, more generally, a set of pairwise coprime integers without $1$) then a necessary and sufficient condition is that $\sum_{p\in A}\frac{1}{p}=\infty$.
The general situation is more complicated. For example s... | 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 |
2,262 | EP-693 | Erdős Problem #693 | Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate $ \max_{i} a_{i+1}-a_i. $ Is this $\leq (\log n)^{O(1)}$?
",
"difficulty": "L1"
},{ | <!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The maximal-gap problem for integers with a divisor in (n,2n) remains open.
**Verified partial progress.**
No distinct partial result was verified beyond t... | 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 |
2,263 | EP-694 | Erdős Problem #694 | Let $f_{\max}(n)$ be the largest $m$ such that $\phi(m)=n$, and $f_{\min}(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function. Investigate $ \max_{n\leq x}\frac{f_{\max}(n)}{f_{\min}(n)}. $ | Carmichael has asked whether there is an integer $n$ for which $\phi(m)=n$ has exactly one solution, that is, $\frac{f_{\max}(n)}{f_{\min}(n)}=1$. Erd\H{o}s has proved that if such an $n$ exists then there must be infinitely many such $n$.",
"difficulty": "L1"
},{
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature rev... | 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 694 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/694",
"label": "Erdős Problems #694: 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 #694: solved (Lean): https://www.erdosproblems.com/694
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 |
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