Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

A colouring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by chi(CF)(H). Pach and Tardos proved that for an (2r -1)-uniform hypergraph H with m edges, chi(CF)(H) is at most of the order of rm(1/r) log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. 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-colourable.