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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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1,217 Problems · 2 with reviewed evidence

12/26

NumberQuestionOpen
#529No statement retained — open to read what the source holdsopenNo formal declaration
#530No statement retained — open to read what the source holdsopenNo formal declaration
#531No statement retained — open to read what the source holdsopenNo formal declaration
#532If N\mathbb{N} is 2-coloured then is there some infinite set A⊆NA\subseteq \mathbb{N} such that all finite subset sums∑n∈Sn \sum_{n\in S}n(as SS ranges over all non-empty finite subsets of AA) are monochromatic?proved (Lean)Formalized
#533Let δ>0\delta > 0. If nn is sufficiently large and GG is a graph on nn vertices with no K5K_5 and at least δn2\delta n^2 edges, must GG contain a set of ≫δn\gg_\delta n vertices spanning no triangle?disprovedFormalized
#534No statement retained — open to read what the source holdssolvedNo formal declaration
#535Let r≥3r \geq 3, and let fr(N)f_r(N) denote the size of the largest subset of {1,…,N}\{1,\ldots,N\} such that no subset of size rr has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that f3(N)>Nc/log⁡log⁡Nf_3(N) > N^{c/\log\log N} for some constant c>0c > 0, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all r≥3r \geq 3.openFormalized
#536Let ϵ>0\epsilon>0 and NN be sufficiently large. Is it true that if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has size at least ϵN\epsilon N then there must be distinct a,b,c∈Aa,b,c\in A such that [a,b]=[b,c]=[a,c],[a, b]=[b, c]=[a, c], where [⋅,⋅][\cdot, \cdot] denotes the least common multiple?openFormalized
#537Let ϵ>0\epsilon>0 and NN be sufficiently large. If A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has ∣A∣≥ϵN\lvert A\rvert \geq \epsilon N then must there exist a1,a2,a3∈Aa_1,a_2,a_3\in A and distinct primes p1,p2,p3p_1,p_2,p_3 such that a1p1=a2p2=a3p3?a_1p_1=a_2p_2=a_3p_3?disproved (Lean)Formalized
#538Let r≥2r\geq 2 and suppose that A⊆{1,…,N}A\subseteq\{1,\ldots,N\} is such that, for any mm, there are at most rr solutions to m=pam=pa where pp is prime and a∈Aa\in A. Give the best possible upper bound for ∑n∈A1n\sum_{n\in A}\frac{1}{n}.openFormalized
#539Let h(n)h(n) be maximal such that, for any set A⊆NA\subseteq \mathbb{N} of size nn, the set{a(a,b):a,b∈A}\left\{ \frac{a}{(a,b)}: a,b\in A\right\}has size at least h(n)h(n). Estimate h(n)h(n).openFormalized
#540Is it true that if A⊆Z/NZA\subseteq \mathbb{Z}/N\mathbb{Z} has size ≫N1/2\gg N^{1/2} then there exists some non-empty S⊆AS\subseteq A such that ∑n∈Sn≡0(modN)\sum_{n\in S}n\equiv 0\pmod{N}?proved (Lean)Formalized
#541Let a1,…,apa_1, \dots, a_p be (not necessarily distinct) residues modulo a prime pp, such that there exists some rr so that if S⊆[p]S \subseteq [p] is non-empty and ∑i∈Sai≡0(modp)\sum_{i \in S} a_i \equiv 0 \pmod{p} then ∣S∣=r|S| = r.proved (Lean)Formalized
#542No statement retained — open to read what the source holdssolvedNo formal declaration
#543No statement retained — open to read what the source holdsdisprovedNo formal declaration
#544No statement retained — open to read what the source holdsopenNo formal declaration
#545No statement retained — open to read what the source holdsopenNo formal declaration
#546No statement retained — open to read what the source holdsprovedNo formal declaration
#547No statement retained — open to read what the source holdsdecidableNo formal declaration
#548No statement retained — open to read what the source holdsfalsifiableNo formal declaration
#549No statement retained — open to read what the source holdsdisprovedNo formal declaration
#550No statement retained — open to read what the source holdsopenNo formal declaration
#551No statement retained — open to read what the source holdsdecidableNo formal declaration
#552No statement retained — open to read what the source holdsopenNo formal declaration
#553No statement retained — open to read what the source holdsprovedNo formal declaration
#554No statement retained — open to read what the source holdsopenNo formal declaration
#555No statement retained — open to read what the source holdsopenNo formal declaration
#556No statement retained — open to read what the source holdsdecidableNo formal declaration
#557No statement retained — open to read what the source holdsopenNo formal declaration
#558No statement retained — open to read what the source holdsopenNo formal declaration
#559No statement retained — open to read what the source holdsdisprovedNo formal declaration
#560No statement retained — open to read what the source holdsopenNo formal declaration
#561No statement retained — open to read what the source holdsopenNo formal declaration
#562Let Rr(n)R_r(n) denote the rr-uniform hypergraph Ramsey number: the minimal mm such that if we 22-colour all edges of the complete rr-uniform hypergraph on mm vertices then there must be some monochromatic copy of the complete rr-uniform hypergraph on nn vertices.openFormalized
#563No statement retained — open to read what the source holdsopenNo formal declaration
#564Let R3(n)R_3(n) be the minimal mm such that if the edges of the 33-uniform hypergraph on mm vertices are 22-coloured then there is a monochromatic copy of the complete 33-uniform hypergraph on nn vertices.openFormalized
#565No statement retained — open to read what the source holdsprovedNo formal declaration
#566Let GG be such that any subgraph on kk vertices has at most 2k−32k-3 edges. Is it true that, if HH has mm edges and no isolated vertices, then r^(G,H)≪m\hat{r}(G,H) \ll m?openFormalized
#567Erdős Problem 567 (Q3)openFormalized
#568No statement retained — open to read what the source holdsopenNo formal declaration
#569No statement retained — open to read what the source holdsopenNo formal declaration
#570No statement retained — open to read what the source holdsprovedNo formal declaration
#571No statement retained — open to read what the source holdsopenNo formal declaration
#572No statement retained — open to read what the source holdsopenNo formal declaration
#573No statement retained — open to read what the source holdsopenNo formal declaration
#574No statement retained — open to read what the source holdsdisprovedNo formal declaration
#575No statement retained — open to read what the source holdsopenNo formal declaration
#576No statement retained — open to read what the source holdsopenNo formal declaration

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