On infinite sets with no $3$ on a line

TL;DR

This paper constructs an infinite point set in the plane where every subset contains a dense, three-collinear-free subset, yet the entire set cannot be finitely partitioned into such subsets, answering a question by Erdős, N"{e}setr"{i}l, and R"{o}dl.

Contribution

It provides a novel construction of an infinite point set with specific collinearity properties, offering a new proof of a longstanding question.

Findings

Every subset has a dense three-collinear-free subset.

The entire set cannot be finitely partitioned into collinearity-free subsets.

The construction answers a question posed by Erdős, N"{e}setr"{i}l, and R"{o}dl.

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.