@article{19017,
  abstract     = {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, Erdős and Sós [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, Erdős and Sós, for k=3 by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for k=4 by Glock, Joos, Kim, Kühn, 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-Erdős-Sós 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.},
  author       = {Glock, Stefan and Kim, Jaehoon and Lichev, Lyuben and Pikhurko, Oleg and Sun, Shumin},
  issn         = {1496-4279},
  journal      = {Canadian Journal of Mathematics},
  pages        = {1--43},
  publisher    = {Cambridge University Press},
  title        = {{On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7}},
  doi          = {10.4153/s0008414x25000021},
  year         = {2025},
}