Lines induced by bichromatic point sets

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 \(Cr(2n−r)\).