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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

4/26

NumberQuestionOpen
#145Let s1<s2<⋯s_1 < s_2 < \cdots be the sequence of squarefree numbers. Is it true that, for any α≥0\alpha\geq 0, lim⁡x→∞1x∑sn≤x(sn+1−sn)α \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha exists?openFormalized
#146If HH is bipartite and is rr-degenerate, that is, every induced subgraph of HH has minimum degree ≤r\leq r, then ex(n;H)≪n2−1/r.\mathrm{ex}(n;H) \ll n^{2-1/r}.open (Lean)Formalized
#147No statement retained — open to read what the source holdsdisprovedNo formal declaration
#148No statement retained — open to read what the source holdsopenNo formal declaration
#149No statement retained — open to read what the source holdsopenNo formal declaration
#150A minimal cut of a graph is a minimal set of vertices whose removal disconnects the graph. Let c(n)c(n) be the maximum number of minimal cuts a graph on nn vertices can have.proved (Lean)Formalized
#151No statement retained — open to read what the source holdsopenNo formal declaration
#152Must lim f n = ∞?provedFormalized
#153Let AA be a finite Sidon set and A+A={s1<⋯<st}A+A=\{s_1<\cdots<s_t\}. Is it true that 1t∑1≤i<t(si+1−si)2→∞\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty as ∣A∣→∞\lvert A\rvert\to \infty?openFormalized
#154Let A⊂{1,…,N}A\subset \{1,\ldots,N\} be a Sidon set with ∣A∣∼N1/2\lvert A\rvert\sim N^{1/2}. Must A+AA+A be well-distributed over all small moduli? In particular, must about half the elements of A+AA+A be even and half odd?proved (Lean)Formalized
#155Is it true that for every k≥1k \geq 1 we have F(N+k)≤F(N)+1 F(N + k) \leq F(N) + 1 for all sufficiently large NN?openFormalized
#156Does there exist a maximal Sidon set A⊂{1,…,N}A\subset \{1,\ldots,N\} of size O(N1/3)O(N^{1/3})?openFormalized
#157No statement retained — open to read what the source holdsprovedNo formal declaration
#158A set is B₂[1] iff it is Sidon.openFormalized
#159No statement retained — open to read what the source holdsopenNo formal declaration
#160Estimate h(n)h(n) by finding a better lower bound.openFormalized
#161No statement retained — open to read what the source holdsopenNo formal declaration
#162No statement retained — open to read what the source holdsopenNo formal declaration
#163No statement retained — open to read what the source holdsprovedNo formal declaration
#164A set A⊂NA\subset \mathbb{N} is primitive if no member of AA divides another. Is the sum ∑n∈A1nlog⁡n\sum_{n\in A}\frac{1}{n\log n} maximised over all primitive sets when AA is the set of primes?proved (Lean)Formalized
#165No statement retained — open to read what the source holdsopenNo formal declaration
#166No statement retained — open to read what the source holdsprovedNo formal declaration
#167No statement retained — open to read what the source holdsfalsifiableNo formal declaration
#168Sanity check: if S is a maximal non ternary subset of {1,..., N} then F N is given by the cardinality of SopenFormalized
#169No statement retained — open to read what the source holdsopenNo formal declaration
#170The problem is to determine the limit of the sequence F(N)N\frac{F(N)}{\sqrt{N}} as N→∞N \to \infty.openFormalized
#171No statement retained — open to read what the source holdsprovedNo formal declaration
#172Is it true that in any finite colouring of N\mathbb{N} there exist arbitrarily large finite AA such that all sums and products of distinct elements in AA are the same colour?openFormalized
#173No statement retained — open to read what the source holdsopenNo formal declaration
#174No statement retained — open to read what the source holdsopenNo formal declaration
#175No statement retained — open to read what the source holdsprovedNo formal declaration
#176No statement retained — open to read what the source holdsopenNo formal declaration
#177No statement retained — open to read what the source holdsopenNo formal declaration
#178Let A1,A2,…A_1,A_2,\ldots be an infinite collection of infinite sets of integers, say Ai={ai1<ai2<⋯ }A_i=\{a_{i1}<a_{i2}<\cdots\}. Does there exist some f:N→{−1,1}f:\mathbb{N}\to\{-1,1\} such that max⁡m,1≤i≤d∣∑1≤j≤mf(aij)∣≪d1\max_{m, 1\leq i\leq d} \left\lvert \sum_{1\leq j\leq m} f(a_{ij})\right\rvert \ll_d 1 for all d≥1d\geq 1?proved (Lean)Formalized
#179No statement retained — open to read what the source holdsprovedNo formal declaration
#180If F\mathcal{F} is a finite set of finite graphs then ex(n;F)\mathrm{ex}(n;\mathcal{F}) is the maximum number of edges a graph on nn vertices can have without containing any subgraphs from F\mathcal{F}. Note that it is trivial that ex(n;F)≤ex(n;G)\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G) for every G∈FG\in\mathcal{F}. Is it true that, for every F\mathcal{F}, there exists G∈FG\in\mathcal{F} such that ex(n;G)≪Fex(n;F)?\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?open (Lean)Formalized
#181No statement retained — open to read what the source holdsopenNo formal declaration
#182No statement retained — open to read what the source holdsprovedNo formal declaration
#183Let R(3;k)R(3;k) be the minimal nn such that if the edges of KnK_n are coloured with kk colours then there must exist a monochromatic triangle. Determine lim⁡k→∞R(3;k)1/k.\lim_{k\to \infty}R(3;k)^{1/k}.open (Lean)Formalized
#184Any graph on nn vertices can be decomposed into O(n)O(n) many edge-disjoint cycles and edges.openFormalized
#185No statement retained — open to read what the source holdsprovedNo formal declaration
#186No statement retained — open to read what the source holdssolvedNo formal declaration
#187No statement retained — open to read what the source holdsopenNo formal declaration
#188What is the smallest kk such that R2\mathbb{R}^2 can be red/blue coloured with no pair of red points unit distance apart, and no kk-term arithmetic progression of blue points with distance 1?openFormalized
#189If R2\mathbb{R}^2 is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?disproved (Lean)Formalized
#190No statement retained — open to read what the source holdssolvedNo formal declaration
#191No statement retained — open to read what the source holdsprovedNo formal declaration
#192No statement retained — open to read what the source holdssolved (Lean)No formal declaration

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