  We show that the maximum number of triples on $n$~points, if no three triples
span at most five points, is $(1\pm o(1))n^2/5$. 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 $\lim_{n\to
\infty}n^{-2}f^{(3)}(n;k,k-2)$ 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~$K_n$ for a suitably defined graph~$H$.
