Coloring points with respect to squares
Eyal Ackerman, Bal\'azs Keszegh, M\'at\'e Vizer

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