Coloring points with respect to squares

TL;DR

This paper proves that for any finite set of points in the plane, there exists a polynomial-time method to 2-color points so that large enough axis-aligned squares contain points of both colors, extending to parallelograms via affine transformations.

Contribution

It establishes a constant threshold for 2-coloring points to ensure mixed colors in large axis-parallel squares with a constructive polynomial-time algorithm.

Findings

Existence of a constant m for 2-coloring points with respect to squares

Polynomial-time algorithm for the 2-coloring problem

Extension of results to homothets of parallelograms

Abstract

We consider the problem of $2$-coloring geometric hypergraphs. Specifically, we show that there is a constant $m$ such that any finite set of points in the plane $\mathcal{S} \subset {\mathbb R}^2$ can be $2$-colored such that every axis-parallel square that contains at least $m$ points from $\mathcal{S}$ contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a $2$-coloring. By affine transformations this result immediately applies also when considering $2$-coloring points with respect to homothets of a fixed parallelogram.