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arXiv · 2602.21275

On infinite sets with no $3$ on a line

Moe Putterman·Mehtaab Sawhney·Gregory Valiant
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Abstract

We give a construction of an infinite set of points $A$ in $\mathbb{R}^2$ such that any subset $P\subseteq A$ has a constant density subset $P'$ with no three points collinear and yet $A$ cannot be separated into finitely many subsets such that each subset has no three points collinear. This provides a new proof of a question of Erdős, Nešetřil, and Rödl. The construction was generated by an internal model at OpenAI.

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Moe Putterman, Mehtaab Sawhney, Gregory Valiant. 2026-02-24. On infinite sets with no $3$ on a line. https://arxiv.org/abs/2602.21275

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