A counterexample to a geometric Hales-Jewett type conjecture

Abstract

Pór 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 $k,l\ge 3$.

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