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Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniformhypergraph on $n$ vertices not containing a subgraph with $k$ edges and at most$s$ vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit$$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all $k$ and confirmedit for $k=2$. Recently, Glock showed this for $k=3$. We settle the next opencase, $k=4$, by showing that $f^{(3)}(n;6,4)=\left(\frac{7}{36}+o(1)\right)n^2$as $n\to\infty$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in[2,r-1]$, we compute the value of the limit $\lim_{n\to \infty}n^{-t}f^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.

On the $(6,4)$-problem of Brown, Erd\H{o}s and S\'os