Bichromatic lines in the plane

TL;DR

This paper proves a conjecture that in a set of red and blue points with no monochromatic line, there are at least |P|-1 bichromatic lines, even under weaker conditions than previously assumed.

Contribution

It establishes the minimum number of bichromatic lines in a two-colored point set under broader conditions than earlier results.

Findings

At least |P|-1 bichromatic lines exist in the specified point set.

The result holds even if the points are not in general position or form a near-pencil.

The proof confirms a conjecture by Kleitman and Pinchasi from 2003.

Abstract

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 We prove the following conjecture of Kleitman and Pinchasi (unpublished, 2003).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.