Coloring Points with Respect to Squares

We consider the problem of $2$-coloring geometric hypergraphs. Specifically, we show that there is a constant $m$ such that any finite set $\mathcal{S}$ of points in the plane 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 homothets of a fixed parallelogram.

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