Lines induced by bichromatic point sets

TL;DR

This paper extends Beck's theorem to bichromatic point sets, establishing lower bounds on the number of lines spanned by points of different colors, with implications for billiard orbit studies.

Contribution

It provides a new bichromatic version of Beck's theorem, quantifying the minimum number of lines spanned by differently colored points under collinearity constraints.

Findings

Establishes a lower bound for lines spanned by bichromatic points.

Generalizes Beck's theorem to bichromatic point configurations.

Applicable to billiard orbit analysis in geometric contexts.

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).