Hitting sets and colorings of hypergraphs
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 -coloring of a vertex set such that every hyperedge contains a vertex of all color classes. We also investigate the connection of this problem with -shallow hitting sets: sets of vertices that intersect each hyperedge in at least one and at most vertices. We determine for some hypergraph families the minimal for which a -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.
Cite
@article{arxiv.2307.12154,
title = {Hitting sets and colorings of hypergraphs},
author = {Balázs Bursics and Bence Csonka and Luca Szepessy},
journal= {arXiv preprint arXiv:2307.12154},
year = {2026}
}