Note on polychromatic coloring of hereditary hypergraph families II

TL;DR

This paper extends constructions of hypergraphs with specific coloring properties, providing new examples for all uniformities of the form 2h-1 for h≥3, using probabilistic and computational methods.

Contribution

It introduces a new family of hypergraphs with no polychromatic 3-coloring but with all h-heavy subhypergraphs 2-colorable, for all h≥3.

Findings

Constructed hypergraphs for all h≥3 with specified coloring properties.

Proved existence for h≥9 using probabilistic methods.

Verified cases 4≤h≤8 through exhaustive computer checks.

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.