Polychromatic Colorings of Unions of Geometric Hypergraphs

TL;DR

This paper investigates polychromatic coloring problems for unions of geometric hypergraphs, demonstrating limitations and providing explicit constructions of non-2-colorable hypergraphs with axis-parallel rectangles.

Contribution

It extends previous results by showing unions of certain geometric hypergraphs cannot always be polychromatically colored and constructs explicit non-2-colorable hypergraphs with large uniformity.

Findings

Union of bottomless rectangles and horizontal strips not always polychromatically colorable

Provides explicit non-2-colorable hypergraphs with axis-parallel rectangles of large uniformity

Strengthens previous results for axis-aligned rectangles

Abstract

We consider the polychromatic coloring problems for unions of two or more geometric hypergraphs on the same vertex sets of points in the plane. We show, inter alia, that the union of bottomless rectangles and horizontal strips does in general not allow for polychromatic colorings. This strengthens the corresponding result of Chen, Pach, Szegedy, and Tardos [Random Struct. Algorithms, 34:11-23, 2009] for axis-aligned rectangles, and gives the first explicit (not randomized) construction of non-$2$-colorable hypergraphs defined by axis-parallel rectangles of arbitrarily large uniformity.