A note on rainbow-free colorings of uniform hypergraphs

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 ZivnATIN SMALL LETTER Y WITH ACUTE showed that p = (k-1)(log n)/n is the threshold function for the existence of a rainbowfree 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.(c) 2023 Elsevier B.V. All rights reserved.