High girth hypergraphs with unavoidable monochromatic or rainbow edges

A classical result of Erdős 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.

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