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Erdős #201 · additive combinatorics / arithmetic progressions

Let Gk(N)G_k(N) be such that any set of NN integers contains a subset of size at least Gk(N)G_k(N) which does not contain a kk-term arithmetic progression. Determine the size of Gk(N)G_k(N). How does it relate to Rk(N)R_k(N), the …

MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let Gk(N)G_k(N) be such that any set of NN integers contains a subset of size at least Gk(N)G_k(N) which does not contain a kk-term arithmetic progression. Determine the size of Gk(N)G_k(N). How does it relate to Rk(N)R_k(N), the size of the largest subset of {1,…,N}\{1,\ldots,N\} without a kk-term arithmetic progression? Is it true that

lim⁡N→∞R3(N)G3(N)=1?\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?

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Background

First asked and investigated by Riddell [Ri69]. It is trivial that Gk(N)≤Rk(N)G_k(N)\leq R_k(N), and it is possible that Gk(N)<Rk(N)G_k(N) <R_k(N) (for example G3(5)=3G_3(5)=3 and R3(5)=4R_3(5)=4, and G3(14)≤7G_3(14)\leq 7 and R3(14)=8R_3(14)=8).

Komlós, Sulyok, and Szemerédi [KSS75] have shown that Rk(N)≪kGk(N)R_k(N) \ll_k G_k(N).

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #201 link

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