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 $(1\pm o(1))n^2\mathrm{/}5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices not containing a subgraph with $k$~vertices and $s$~edges. In 1973, Brown, Erd\H{o}s and S'os conjectured that the limit $\lim_{n\to \infty}n^{-2}f^{(3)}(n;k,k-2)$existsforall$k$\mathrm{.}Theyprovedthisfor$k=4$, where the limit is $1\mathrm{/}6$ and the extremal examples are Steiner triple systems. We prove the conjecture for $k=5$ and show that the limit is~$1\mathrm{/}5$. The upper bound is established via a simple optimisation problem. For the lower bound, we use approximate $H$-decompositions of~$K_n$ for a suitably defined graph~$H$.