Triple systems with no three triples spanning at most five points

We show that the maximum number of triples on n points, if no three triples span at most five points, is (1 +/- o(1))n2/5. More generally, let f(r)(n;k,s) be the maximum number of edges in an r-uniform hypergraph on n vertices not containing a subgraph with k vertices and s edges. In 1973, Brown, Erdos and Sos conjectured that the limit limn ->infinity n-2f(3)(n;k,k-2) exists for all k. They proved this for k=4, where the limit is 1/6 and the extremal examples are Steiner triple systems. We prove the conjecture for k=5 and show that the limit is 1/5. The upper bound is established via a simple optimisation problem. For the lower bound, we use approximate H-decompositions of Kn for a suitably defined graph H.