On a problem of Brown, Erd\H{o}s and S\'{o}s

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph not containing a subhypergraph with $k$ edges on at most $s$ vertices. Recently, Delcourt and Postle, building on work of Glock, Joos, Kim, K\"{u}hn, Lichev and Pikhurko, proved that the limit $\lim_{n \to \infty} n^{-2} f^{(3)}(n;k+2,k)$ exists for all $k \ge 2$, solving an old problem of Brown, Erd\H{o}s and S\'{o}s (1973). Meanwhile, Shangguan and Tamo asked the more general question of determining if the limit $\lim_{n \to \infty} n^{-t} f^{(r)}(n;k(r-t)+t,k)$ exists for all $r>t\ge 2$ and $k \ge 2$. Here we make progress on their question. For every even $k$, we determine the value of the limit when $r$ is sufficiently large with respect to $k$ and $t$. Moreover, we show that the limit exists for $k \in \{5,7\}$ and all $r > t \ge 2$.