Coloring Questions on Axis-Parallel Rectangles and Arithmetic Progressions

TL;DR

This paper constructs explicit hypergraphs with large chromatic number and uniformity, providing new geometric and number-theoretic insights, including a deterministic proof of a rectangle coloring theorem and applications to integer progressions and posets.

Contribution

It offers explicit constructions of hypergraphs with large chromatic number, replacing probabilistic methods, and extends results to geometric, number-theoretic, and order-theoretic contexts.

Findings

Explicit hypergraph constructions with large chromatic number

Deterministic proof of rectangle coloring theorem

Realization of graphs with large girth and chromatic number

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.