A classical result of Erdős and Hajnal claims that for any integers k,r,g≥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
Nearby in the stack
colors there is a monochromatic hyperedge. We show that for any integers
r,g≥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.