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On infinite sets with no $3$ on a line

Moe Putterman, Mehtaab Sawhney, Gregory Valiant

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.

On infinite sets with no $3$ on a line

Abstract

We give a construction of an infinite set of points in such that any subset has a constant density subset with no three points collinear and yet 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.
Paper Structure (3 sections, 1 theorem, 3 equations)

This paper contains 3 sections, 1 theorem, 3 equations.

Table of Contents

  1. Introduction
  2. Comment on use of AI
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

There exists an infinite set $A\subseteq \mathbb{R}^{2}$ with the following properties:

Theorems & Definitions (4)

  • Theorem 1.1
  • Claim 2.1
  • proof
  • Remark