Asymptotics of the Brown--Erdős--Sós problem at integer exponents

The Brown--Erdős--Sós problem is a fundamental problem in sparse hypergraph Turán theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In