Lines induced by bichromatic point sets

Abstract

An important theorem of Beck says that any point set in the Euclidean plane is either ``nearly general position'' or ``nearly collinear'': there is a constant C>0 such that, given n points in the plane with at most r$ of them collinear, the number of lines induced by the points is at least Cr(n-r). Recent work of Gutkin-Rams on billiards orbits requires the following elaboration of Beck's Theorem to bichromatic point sets: there is a constant C>0 such that, given n red points and n blue points in the plane with at most r of them collinear, the number of lines spanning at least one point of each color is at least Cr(2n-r).