Problem 846

Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\epsilon n$ with no three on a line.

Is it true that $A$ is the union of a finite number of sets where no three are on a line?

Source: erdosproblems.com/846

Geometry

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SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The site credits two independent disproofs, by DeepMind and by an internal model at OpenAI, the latter written up by Putterman, Sawhney and Valiant [PSV26]; the claim pages Putterman, Sawhney and Valiant and DeepMind record the two results and their acceptance.

Er92b: Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
PSV26: M. Putterman, M. Sawhney, and G. Valiant, On infinite sets with no $3$ on a line. arXiv:2602.21275 (2026).
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.

Problem 846

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