A note on rainbow-free colorings of uniform hypergraphs

Abstract

A rainbow-free coloring of a k-uniform hypergraph H is a vertex-coloring which uses k colors but with the property that no edge of H attains all colors. Koerkamp and Z̆ivný showed that p=(k−1)(logn)/n is the threshold function for the existence of a rainbow-free coloring of the random k-uniform hypergraph Gk(n,p), and presented that the case when p is close to the threshold is open. In this paper, we give an answer to the question.