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Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

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  • Open per source608
  • Resolved per source556
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Exact formal statement available604 / 1,217
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1,217 Problems · 2 with reviewed evidence

16/26

NumberQuestionOpen
#721No statement retained — open to read what the source holdssolvedNo formal declaration
#722No statement retained — open to read what the source holdsprovedNo formal declaration
#723If there is a finite projective plane of order nn then must nn be a prime power?falsifiableFormalized
#724No statement retained — open to read what the source holdsopenNo formal declaration
#725No statement retained — open to read what the source holdsopenNo formal declaration
#726As n→∞n\to \infty ranges over integers ∑p≤n1n∈(p/2,p)(modp)1p∼log⁡log⁡n2\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}?openFormalized
#727Let k≥2k ≥ 2. Does ((n+k)!)2∣(2n)!((n+k)!)^2∣(2n)! hold for infinitely many nn?openFormalized
#728Let ε\varepsilon be sufficiently small and C,C′>0C, C' > 0. Are there integers a,b,na, b, n such that a,b>εna! b!∣n! (a+b−n)!,a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!, and Clog⁡n<a+b−n<C′log⁡n?C \log n < a + b - n < C' \log n ?proved (Lean)Formalized
#729Let C>0C>0 be a constant. Are there infinitely many integers a,b,na,b,n with a+b>n+Clog⁡na+b> n+C\log n such that the denominator of n!a!b!\frac{n!}{a!b!}contains only primes ≪C1\ll_C 1?proved (Lean)Formalized
#730Are there infinitely many pairs of integers n<mn < m such that (2nn)\binom{2n}{n} and (2mm)\binom{2m}{m} have the same set of prime divisors?openFormalized
#731No statement retained — open to read what the source holdsopenNo formal declaration
#732No statement retained — open to read what the source holdsprovedNo formal declaration
#733No statement retained — open to read what the source holdsprovedNo formal declaration
#734No statement retained — open to read what the source holdsopenNo formal declaration
#735No statement retained — open to read what the source holdssolvedNo formal declaration
#736No statement retained — open to read what the source holdsnot provableNo formal declaration
#737No statement retained — open to read what the source holdsprovedNo formal declaration
#738No statement retained — open to read what the source holdsopenNo formal declaration
#739No statement retained — open to read what the source holdsnot provableNo formal declaration
#740Let m\mathfrak{m} be an infinite cardinal and GG be a graph with chromatic number m\mathfrak{m}. Let r≥1r\geq 1. Must GG contain a subgraph of chromatic number m\mathfrak{m} which does not contain any odd cycle of length ≤r\leq r?openFormalized
#741Let A⊆NA\subseteq \mathbb{N} be such that A+AA+A has positive upper density. Can one always decompose A=A1⊔A2A=A_1\sqcup A_2 such that A1+A1A_1+A_1 and A2+A2A_2+A_2 both have positive upper density?solved (Lean)Formalized
#742The complete bipartite graph Ka,bK_{a, b} has exactly a⋅ba \cdot b edges. The bound ⌊n2/4⌋\lfloor n^2 / 4 \rfloor in the Murty-Simon conjecture is attained by the balanced case K⌈n/2⌉,⌊n/2⌋K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}.decidableFormalized
#743No statement retained — open to read what the source holdsfalsifiableNo formal declaration
#744No statement retained — open to read what the source holdsdisprovedNo formal declaration
#745No statement retained — open to read what the source holdsprovedNo formal declaration
#746No statement retained — open to read what the source holdsprovedNo formal declaration
#747No statement retained — open to read what the source holdssolvedNo formal declaration
#748No statement retained — open to read what the source holdsprovedNo formal declaration
#749Let ϵ>0\epsilon>0. Does there exist A⊆NA\subseteq \mathbb{N} such that the lower density of A+AA+A is at least 1−ϵ1-\epsilon and yet 1A∗1A(n)≪ϵ11_A\ast 1_A(n) \ll_\epsilon 1 for all nn?openFormalized
#750Let f(m)f(m) be some function such that f(m)→∞f(m)\to \infty as m→∞m\to \infty. Does there exist a graph GG of infinite chromatic number such that every subgraph on mm vertices contains an independent set of size at least m2−f(m)\frac{m}{2}-f(m)?proved (Lean)Formalized
#751Let GG be a graph with chromatic number χ(G)=4\chi(G)=4. If m1<m2<⋯m_1<m_2<\cdots are the lengths of the cycles in GG then can min⁡(mi+1−mi)\min(m_{i+1}-m_i) be arbitrarily large?disproved (Lean)Formalized
#752No statement retained — open to read what the source holdsprovedNo formal declaration
#753The list chromatic number χL(G)\chi_L(G) is defined to be the minimal kk such that for any assignment of a list of kk colours to each vertex of GG (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.disproved (Lean)Formalized
#754No statement retained — open to read what the source holdsprovedNo formal declaration
#755Erdős asked whether every nn-point set in R6\mathbb{R}^6 spans at most (1/27+o(1))n3(1/27 + o(1)) n^3 unit equilateral triangles.provedFormalized
#756Let A⊂R2A\subset \mathbb{R}^2 be a set of nn points. Can there be ≫n\gg n many distinct distances each of which occurs for more than nn many pairs from AA?proved (Lean)Formalized
#757What is the supremum of the set of admissible numbers?openFormalized
#758No statement retained — open to read what the source holdssolvedNo formal declaration
#759No statement retained — open to read what the source holdssolvedNo formal declaration
#760The cochromatic number of GG, denoted by ζ(G)\zeta(G), is the minimum number of colours needed to colour the vertices of GG such that each colour class induces either a complete graph or independent set.proved (Lean)Formalized
#761No statement retained — open to read what the source holdsopenNo formal declaration
#762The cochromatic number of GG, denoted by ζ(G)\zeta(G), is the minimum number of colours needed to colour the vertices of GG such that each colour class induces either a complete graph or empty graph.disproved (Lean)Formalized
#763No statement retained — open to read what the source holdsdisprovedNo formal declaration
#764No statement retained — open to read what the source holdsdisprovedNo formal declaration
#765No statement retained — open to read what the source holdssolved (Lean)No formal declaration
#766No statement retained — open to read what the source holdsopenNo formal declaration
#767No statement retained — open to read what the source holdsprovedNo formal declaration
#768No statement retained — open to read what the source holdsopenNo formal declaration

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