Coloring Questions on Axis-Parallel Rectangles and Arithmetic Progressions

Abstract

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach, Szegedy, and Tardos. They showed that for any constants $c,k\ge1$, there exists a finite point set $P$ in the plane with the following property: for every coloring of $P$ with $c$ colors, there is an axis-parallel rectangle containing at least $k$ points, all of the same color. Their original proof is probabilistic; we present an explicit construction. Moreover, in the case $k=2$, we show that one can even realize a graph that has arbitrarily large girth and chromatic number simultaneously. We also answer a question of P\'alv\"olgyi on coloring sets of integers with respect to certain finite arithmetic progressions. Finally, we give an application to coloring partially ordered sets.