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

We consider a new type of extremal hypergraph problem: given an r -graph F and an integer k ≥ 2 determine the maximum number of edges in an F -free, k -colourable r -graph on n vertices. tivation 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 0.32975 < 1 3 − 1 280 , improving previous bounds of 1 3 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 1 3 − 4.5305 × 1 0 − 6 due to Mubayi [D. Mubayi, On hypergraphs with every four points spanning at most two triples, Electron. J. Combin. 10 (10) (2003)].