Bichromatic Lines in the Plane

Abstract

Given a set of red and blue points in the plane, a bichromatic line is a line containing at least one red and one blue point. We prove the following conjecture of Kleitman and Pinchasi. Let $P$ be a set of $n$ red, and $n$ or $n-1$ blue points in the plane. If neither color class is collinear, then $P$ determines at least $|P|-1$ bichromatic lines. In fact we are able to achieve the same conclusion under the weaker assumption that $P$ is not collinear or a near-pencil.