Hitting sets and colorings of hypergraphs

TL;DR

This paper explores the minimal edge sizes needed for polychromatic colorings in hypergraphs, investigates shallow hitting sets, and examines their connections with geometric hypergraphs and arithmetic progressions.

Contribution

It introduces new bounds and relations between hypergraph coloring, shallow hitting sets, and geometric hypergraph families, including those induced by arithmetic progressions.

Findings

Determined minimal c for c-shallow hitting sets in certain hypergraph families.

Established connections between geometric hypergraphs and arithmetic progressions.

Proved relations between difference sets and polychromatic colorability.

Abstract

In this paper we study the minimal size of edges in hypergraph families that guarantees the existence of a polychromatic coloring, that is, a $k$-coloring of a vertex set such that every hyperedge contains a vertex of all $k$ color classes. We also investigate the connection of this problem with $c$-shallow hitting sets: sets of vertices that intersect each hyperedge in at least one and at most $c$ vertices. We determine for some hypergraph families the minimal $c$ for which a $c$-shallow hitting set exists. We also study this problem for a special hypergraph family, which is induced by arithmetic progressions with a difference from a given set. We show connections between some geometric hypergraph families and the latter, and prove relations between the set of differences and polychromatic colorability.