A counterexample to a geometric Hales-Jewett type conjecture

TL;DR

This paper disproves a geometric coloring conjecture by providing a counterexample for cases where both the number of colors and the maximum collinear points are at least three, challenging previous assumptions in combinatorial geometry.

Contribution

The paper presents the first known counterexample to a conjecture relating to colored point sets and collinearity in the plane for k, l ≥ 3.

Findings

The conjecture is false for all k, l ≥ 3.

Counterexamples exist that violate the conjecture.

The conjecture holds only for specific cases like l=2 or k=2.

Abstract

P\'or and Wood conjectured that for all $k,l \ge 2$ there exists $n \ge 2$ with the following property: whenever $n$ points, no $l + 1$ of which are collinear, are chosen in the plane and each of them is assigned one of $k$ colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for $l = 2$ (by the pigeonhole principle) and in the case $k = 2$ it is an immediate corollary of the Motzkin-Rabin theorem. In this note we show that the conjecture is false for $k, l \ge 3$.