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, Erdős and Sós conjectured that the limit limn→∞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. [ABSTRACT FROM AUTHOR]