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 _χCF(H). Pach and Tardos proved that for an (2 _r ? 1)-uniform hypergraph H with m edges, _χCF(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.