A solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane

Abstract

Let P be a set of n blue points in the plane, not all on a line. Let R be a set of m red points such that P ∩ R = ∅ and every line determined by P contains a point from R. We provide an answer to an old problem by Grunbaum and Motzkin [9] and independently by Erdős and Purdy [6] who asked how large must m be in terms of n in such a case? More specifically, both [9] and [6] were looking for the best absolute constant c such that m ≥ cn. We provide an answer to this problem and show that m ≥ (n−1)/3.