Bichromatic lines in the plane

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 We prove the following conjecture of Kleitman and Pinchasi (unpublished, 2003). Let P be a set of n red, and n or n-1 blue points in the plane. If neither colour 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.