On infinite sets with no $3$ on a line

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 Erd\H{o}s, Ne\v{s}et\v{r}il, and R\"{o}dl. The construction was generated by an internal model at OpenAI.