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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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1,217 Problems · 2 with reviewed evidence

17/26

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#769Let c(n)c(n) be minimal such that if k≥c(n)k\geq c(n) then the nn-dimensional unit cube can be decomposed into kk homothetic nn-dimensional cubes. Give good bounds for c(n)c(n) — in particular, is it true that c(n)≫nnc(n)\gg n^n?Source says openFormalized
#770n + 1 is prime iff h n = n + 1. This is described as 'easy to see' in [Er74b].Source says openFormalized
#771No statement retained — open to read what the source holdsSource says provedNo formal declaration
#772No statement retained — open to read what the source holdsSource says provedNo formal declaration
#773No statement retained — open to read what the source holdsSource says openNo formal declaration
#774Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?Source says openFormalized
#775Is there a 33-uniform hypergraph on nn vertices which contains at least n−O(1)n-O(1) different sizes of cliques (maximal complete subgraphs)?Source says disproved (Lean)Formalized
#776No statement retained — open to read what the source holdsSource says openNo formal declaration
#777No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#778No statement retained — open to read what the source holdsSource says openNo formal declaration
#779A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810Source says falsifiableFormalized
#780No statement retained — open to read what the source holdsSource says provedNo formal declaration
#781No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#782No statement retained — open to read what the source holdsSource says openNo formal declaration
#783No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#784No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#785Let A,B⊆NA,B\subseteq \mathbb{N} be infinite sets such that A+BA+B contains all large integers. Let A(x)=∣A∩[1,x]∣A(x)=\lvert A\cap [1,x]\rvert and similarly for B(x)B(x). Is it true that if A(x)B(x)∼xA(x)B(x)\sim x then A(x)B(x)−x→∞A(x)B(x)-x\to \infty as x→∞x\to \infty?Source says proved (Lean)Formalized
#786Let ϵ>0\epsilon > 0. Is there some set A⊂NA\subset\mathbb{N} of density >1−ϵ> 1 - \epsilon such that a1⋯ar=b1⋯bsa_1\cdots a_r = b_1\cdots b_s with ai,bj∈Aa_i, b_j\in A can only hold when r=sr = s?Source says openFormalized
#787No statement retained — open to read what the source holdsSource says openNo formal declaration
#788No statement retained — open to read what the source holdsSource says openNo formal declaration
#789Let h(n)h(n) be maximal such that if A⊆ZA\subseteq \mathbb{Z} with ∣A∣=n\lvert A\rvert=n then there is B⊆AB\subseteq A with ∣B∣≥h(n)\lvert B\rvert \geq h(n) such that if a1+⋯+ar=b1+⋯+bsa_1+\cdots+a_r=b_1+\cdots+b_s with ai,bi∈Ba_i,b_i\in B then r=sr=s.Source says openFormalized
#790No statement retained — open to read what the source holdsSource says openNo formal declaration
#791No statement retained — open to read what the source holdsSource says openNo formal declaration
#792No statement retained — open to read what the source holdsSource says openNo formal declaration
#793No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration
#794Is it true that every 33-uniform hypergraph on 3n3n vertices with at least n3+1n^3+1 edges must contain either a subgraph on 44 vertices with 33 edges or a subgraph on 55 vertices with 77 edges?Source says disproved (Lean)Formalized
#795No statement retained — open to read what the source holdsSource says provedNo formal declaration
#796Let k≥2k\geq 2 and let gk(n)g_k(n) be the largest possible size of A⊆{1,…,n}A\subseteq \{1,\ldots,n\} such that every mm has <k<k solutions to m=a1a2m=a_1a_2 with a1<a2∈Aa_1<a_2\in A. Is it true that g3(n)=log⁡log⁡nlog⁡nn+(c+o(1))nlog⁡ng_3(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{\log n} for some constant cc?Source says openFormalized
#797No statement retained — open to read what the source holdsSource says provedNo formal declaration
#798Let t(n)t(n) be the minimum number of points in {1,…,n}2\{1,\ldots,n\}^2 such that the (t2)\binom{t}{2} lines determined by these points cover all points in {1,…,n}2\{1,\ldots,n\}^2.Source says proved (Lean)Formalized
#799No statement retained — open to read what the source holdsSource says provedNo formal declaration
#800No statement retained — open to read what the source holdsSource says provedNo formal declaration
#801No statement retained — open to read what the source holdsSource says provedNo formal declaration
#802No statement retained — open to read what the source holdsSource says openNo formal declaration
#803No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#804No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#805No statement retained — open to read what the source holdsSource says openNo formal declaration
#806No statement retained — open to read what the source holdsSource says provedNo formal declaration
#807No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#808No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#809No statement retained — open to read what the source holdsSource says openNo formal declaration
#810No statement retained — open to read what the source holdsSource says openNo formal declaration
#811No statement retained — open to read what the source holdsSource says openNo formal declaration
#812Is it true that R(n+1)R(n)≥1+c\frac{R(n+1)}{R(n)}\geq 1+c for some constant c>0c>0, for all large nn?Source says openFormalized
#813No statement retained — open to read what the source holdsSource says openNo formal declaration
#814No statement retained — open to read what the source holdsSource says provedNo formal declaration
#815No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#816No statement retained — open to read what the source holdsSource says provedNo formal declaration