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Erdős #160 · additive combinatorics / arithmetic progressions

Let h(N)h(N) be the smallest kk such that {1,…,N}\{1,\ldots,N\} can be coloured with kk colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate h(N)h(N).

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 h(N)h(N) be the smallest kk such that {1,…,N}\{1,\ldots,N\} can be coloured with kk colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate h(N)h(N).

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Background

Investigated by Erdős and Freud. This has been discussed on MathOverflow, where LeechLattice shows

h(N)≪N2/3.h(N) \ll N^{2/3}.

In the comments of this site Hunter improves this to

h(N)≪Nlog⁡3log⁡22+o(1)h(N) \ll N^{\frac{\log 3}{\log 22}+o(1)}

(note log⁡3log⁡22≈0.355\frac{\log 3}{\log 22}\approx 0.355).

The observation of Zach Hunter in that question coupled with recent progress on the size of subsets without three-term arithmetic progression (see [BlSi23] which improves slightly on the bounds due to Kelley and Meka [KeMe23]) imply that

h(N)≫exp⁡(c(log⁡N)1/9)h(N) \gg \exp(c(\log N)^{1/9})

for some c>0c>0.

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 #160 link
  2. [2]Upstream Lean 4 statement for Erdős #160 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 160 link

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