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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
  • Other source status53
Exact formal statement available604 / 1,217
With Repository-reviewed evidence2 / 1,217
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Problems

114 Problems

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NumberQuestionOpen
#45Let k≥2k\geq 2. Is there an integer nkn_k such that, if D={1<d<nk:d∣nk}D=\{ 1<d<n_k : d\mid n_k\}, then for any kk-colouring of DD there is a monochromatic subset D′⊆DD'\subseteq D such that ∑d∈D′1d=1\sum_{d\in D'}\frac{1}{d}=1?proved (Lean)Formalized
#46Does every finite colouring of the integers have a monochromatic solution to 1=∑1ni1=\sum \frac{1}{n_i} with 2≤n1<⋯<nk2\leq n_1<\cdots <n_k?proved (Lean)Formalized
#54No statement retained — open to read what the source holdssolvedNo formal declaration
#55No statement retained — open to read what the source holdssolvedNo formal declaration
#70Erdős Problem 70: Let c\mathfrak{c} be the cardinality of the continuum, let β\beta be a countable ordinal, and let 2≤n<ω2 \le n < \omega. Is it true that c→(β,n)23\mathfrak{c} \to (\beta, n)^3_2?openFormalized
#76No statement retained — open to read what the source holdsprovedNo formal declaration
#77No statement retained — open to read what the source holdsopenNo formal declaration
#78No statement retained — open to read what the source holdsopenNo formal declaration
#79No statement retained — open to read what the source holdsprovedNo formal declaration
#80No statement retained — open to read what the source holdsopenNo formal declaration
#87No statement retained — open to read what the source holdsopenNo formal declaration
#88No statement retained — open to read what the source holdsprovedNo formal declaration
#112No statement retained — open to read what the source holdsopenNo formal declaration
#118No statement retained — open to read what the source holdsdisprovedNo formal declaration
#129No statement retained — open to read what the source holdsopenNo formal declaration
#159No statement retained — open to read what the source holdsopenNo formal declaration
#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
#165No statement retained — open to read what the source holdsopenNo formal declaration
#166No 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
#181No statement retained — open to read what the source holdsopenNo 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
#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
#191No statement retained — open to read what the source holdsprovedNo formal declaration
#439No statement retained — open to read what the source holdsprovedNo formal declaration
#474No statement retained — open to read what the source holdsnot provableNo formal declaration
#483No statement retained — open to read what the source holdsopenNo formal declaration
#484Prove that there exists an absolute constant c>0c>0 such that, whenever {1,…,N}\{1,\ldots,N\} is kk-coloured (and NN is large enough depending on kk) then there are at least cNcN many integers in {1,…,N}\{1,\ldots,N\} which are representable as a monochromatic sum (that is, a+ba+b where a,b∈{1,…,N}a,b\in \{1,\ldots,N\} are in the same colour class and a≠ba\neq b).proved (Lean)Formalized
#508The "chromatic number of the plane" is at least 4. This can be proven by considering the [Moser-Spindel graph](https://de.wikipedia.org/wiki/Moser-Spindel) or the [Golomb graph](https://en.wikipedia.org/wiki/Golomb_graph) graph.openFormalized
#518No statement retained — open to read what the source holdsprovedNo 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
#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
#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

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