Problem 847

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claims/: The 1 claim page of Problem 847, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic progression.

Is it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?

Status. DISPROVED (LEAN). The standing is derived from the claim page, accepted on the refereed publication and the site's credit.

Source. erdosproblems.com/847, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #847, https://www.erdosproblems.com/847.

References.

• [RRS24] Reiher, Christian and Rödl, Vojtěch and Sales, Marcelo, Colouring versus density in integers and Hales-Jewett cubes. J. Lond. Math. Soc. (2) (2024), Paper No. e12987, 24.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• reiher_2024_colouring_versus_density_integers_hales_jewett_cubes