Note on polychromatic coloring of hereditary hypergraph families

TL;DR

This paper presents a counterexample in hypergraph coloring, disproving a conjecture and advancing understanding of polychromatic colorings in hereditary hypergraph families, with implications for future research.

Contribution

It provides the first known hypergraph counterexample to a conjecture on polychromatic coloring, highlighting limitations of current methods and connecting to panchromatic colorings.

Findings

Counterexample hypergraph with no 3-coloring

Disproof of Keszegh's conjecture

Limitations of current coloring methods

Abstract

We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings.