On the $(k+2,k)$-problem of Brown, Erd{o}s and S\'os for $k=5,6,7$

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{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 limn→ ∞n⁻²f⁽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{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{o}s-S\'os Problem for k∈ \5,6,7\. More generally, we obtain the value of limn→ ∞n⁻²f⁽ʳ⁾(n;rk-2k+2,k) for all r≥ 3 and k∈ \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.