  Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex
$r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$
vertices. Brown, Erd\H{o}s and S\'os [New directions in the theory of graphs
(Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press
1973] conjectured that the limit $\lim_{n\rightarrow
\infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was
previously determined for $k=2$ in the original paper of Brown, Erd\H{o}s and
S\'os, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for
$k=4$ by Glock, Joos, Kim, K\"uhn, Lichev and Pikhurko [arXiv:2209.14177,
accepted by Proc. Amer. Math. Soc.] while Delcourt and Postle
[arXiv:2210.01105, accepted by Proc. Amer. Math. Soc.] proved the conjecture
(without determining the limiting value).
  In this paper, we determine the value of the limit in the
Brown-Erd\H{o}s-S\'os Problem for $k\in \{5,6,7\}$. More generally, we obtain
the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all
$r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with
recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new
asymptotic values for several generalised Ramsey numbers.
