Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

A coloring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose color does not get repeated in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF (H). Pach and Tardos studied this parameter for graphs and hypergraphs. Among other things, they proved that for an (2r − 1)-uniform hypergraph H with m edges, χCF (H) is of the order of m logm. They also raised the question whether the same result holds for r-uniform hypergraphs. In this paper we show that this is not necessarily true. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colorable.