# Lines induced by bichromatic point sets

 Authors: Louis Theran Preprint: 1101.1488, 2011 Full text: arXiv

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 $Cr(2n−r)$.