Hitting sets and colorings of hypergraphs

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.