High girth hypergraphs with unavoidable monochromatic or rainbow edges

TL;DR

This paper constructs sparse hypergraphs with high girth that guarantee the existence of either monochromatic or rainbow edges under any vertex coloring, extending classical results on hypergraph colorings.

Contribution

It introduces the first known hypergraphs of high girth that force either monochromatic or rainbow edges in any coloring, with both probabilistic and deterministic proofs.

Findings

Existence of high girth hypergraphs with unavoidable monochromatic or rainbow edges

Probabilistic and deterministic constructions provided

Extends classical hypergraph coloring results to include rainbow edges

Abstract

A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that in any coloring of their vertices with at most $k-1$ colors there is a monochromatic hyperedge. We show that for any integers $r, g\geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ such that in any coloring of its vertices there is either a monochromatic or a rainbow (totally multicolored) edge. We give a probabilistic and a deterministic proof of this result.