A counterexample to a geometric Hales-Jewett type conjecture

Pór and Wood conjectured that for all k, l ≥ 2 there exists n ≥ 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 ≥ 3.