Chromatic Turán problems and a new upper bound for the Turán density of

We consider a new type of extremal hypergraph problem: given an r-graph and an integer k≥2 determine the maximum number of edges in an -free, k-colourable r-graph on n vertices.Our motivation for studying such problems is that it allows us to give a new upper bound for an old Turán problem. We show that a 3-graph in which any four points span at most two edges has density less than , improving previous bounds of due to de Caen D. de Caen, Extension of a theorem of Moon and Moser on complete subgraphs, Ars Combin. 16 (1983) 5–10], and due to Mubayi D. Mubayi, On hypergraphs with every four points spanning at most two triples, Electron. J. Combin. 10 (10) (2003)].