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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~$1/5$. 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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~$1/5$. 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