On the (k+2, k) -problem of Brown, Erdos 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, Erdos 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 ₍ 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, Erdos 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-Erdos-S\'os Problem for k \5, 6, 7\. More generally, we obtain the value of ₍ 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.