An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

Abstract

The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k − 1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We could only prove slightly weaker results for dual hypergraphs defined by pseudohalfplanes, and for hypergraphs defined by pseudohalfspheres. We also introduce another new notion that seems to be important for investigating polychromatic colorings and ǫ-nets, shallow hitting sets. We show that all the above hypergraphs have shallow hitting sets, if their hyperedges are containment-free.

Coloring Intersection Hypergraphs of Pseudo-Disks

“…Finally we mention that similarly to pseudo-disks, pseudo-halfplanes are natural generalizations of halfplanes. For pseudo-halfplanes it was also possible to reprove the same coloring results that are known for halfplanes [17]. Also, due to the lack of direct relation to pseudodisks, we did not mention axis-parallel rectangles, whose intersection hypergraph coloring problems are some of the most interesting open problems of the area.…”

Three-chromatic geometric hypergraphs

“…Theorem 7 (Keszegh-Pálvölgyi [20]). Any (2k − 1)-heavy hypergraph realizable by pseudohalfplanes is polychromatic k-colorable, i.e., given a finite set of points and a pseudohalfplane arrangement in the plane, the points can be k-colored such that every pseudohalfplane that contains at least 2k − 1 points contains all k colors.…”

Unsplittable coverings in the plane

“…If the answer to this question is yes, in some sense this could be regarded as an extension of the Lovász local lemma [10]. For more problems and results related to shiftchains and special shift-chains, consult [36,25].…”

“…According to Theorem 6.3, for any m ≥ 3, every m-uniform special shift-chain is 2colorable. Keszegh and Pálvölgyi [25] recently extended this theorem to show that the vertices of every (2k − 1)-uniform special shift-chain can be colored by k colors so that every hyperedge contains at least one point of each color.…”