Note on polychromatic coloring of hereditary hypergraph families II

Abstract

We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy restricted subhypergraphs are $2$-colorable. Together with the previously known case $h=3$, this gives examples with uniformity $2h-1$ for every $h\ge 3$. The construction is based on complements of suitable $h$-uniform hypergraphs on $3h-1$ vertices. For $h\ge 9$ we prove existence by a simple probabilistic argument; the remaining cases $4\le h\le 8$ are certified by a short exhaustive computer check, whose fully reproducible description and source code are included in the appendix.