High girth hypergraphs with unavoidable monochromatic or rainbow edges

Abstract

A classical result of Erdős and Hajnal claims that for any integers $k,r,g\ge 2$ 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 $r,g\ge 2$ 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.

Corrected remark to references. The paper [8] addresses both graphs and hypergraphs