On the $(6,4)$-problem of Brown, Erdős and Sós

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$ vertices not containing a subgraph with $k$ edges and at most $s$ vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\underset{n\to \mathrm{\infty }}{\lim }{n}^{-2}{f}^{(3)}(n;k+2,k)$$ exists for all $k$ and confirmed it for $k=2$. Recently, Glock showed this for $k=3$. We settle the next open case, $k=4$, by showing that $f^{(3)}(n;6,4)=(\frac{7}{36}+o(1))n^2$ as $n\to \mathrm{\infty }$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in [2,r-1]$, we compute the value of the limit ${\lim }_{n\to \mathrm{\infty }}{n}^{-t}{f}^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.