{"OA_place":"publisher","page":"1-43","type":"journal_article","author":[{"first_name":"Stefan","last_name":"Glock","full_name":"Glock, Stefan"},{"last_name":"Kim","full_name":"Kim, Jaehoon","first_name":"Jaehoon"},{"id":"9aa8388e-d003-11ee-8458-c4c1d7447977","full_name":"Lichev, Lyuben","last_name":"Lichev","first_name":"Lyuben"},{"first_name":"Oleg","last_name":"Pikhurko","full_name":"Pikhurko, Oleg"},{"full_name":"Sun, Shumin","last_name":"Sun","first_name":"Shumin"}],"arxiv":1,"month":"01","department":[{"_id":"MaKw"}],"date_created":"2025-02-10T08:39:46Z","scopus_import":"1","_id":"19017","date_published":"2025-01-06T00:00:00Z","oa_version":"Published Version","isi":1,"has_accepted_license":"1","day":"06","title":"On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7","language":[{"iso":"eng"}],"publication":"Canadian Journal of Mathematics","publication_identifier":{"issn":["0008-414X"],"eissn":["1496-4279"]},"abstract":[{"lang":"eng","text":"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).\r\nIn 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."}],"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.4153/s0008414x25000021"}],"ddc":["500"],"publication_status":"epub_ahead","external_id":{"arxiv":["2403.04474"],"isi":["001416788600001"]},"year":"2025","publisher":"Cambridge University Press","article_type":"original","citation":{"ista":"Glock S, Kim J, Lichev L, Pikhurko O, Sun S. 2025. On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. Canadian Journal of Mathematics., 1–43.","ieee":"S. Glock, J. Kim, L. Lichev, O. Pikhurko, and S. Sun, “On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7,” <i>Canadian Journal of Mathematics</i>. Cambridge University Press, pp. 1–43, 2025.","ama":"Glock S, Kim J, Lichev L, Pikhurko O, Sun S. On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. <i>Canadian Journal of Mathematics</i>. 2025:1-43. doi:<a href=\"https://doi.org/10.4153/s0008414x25000021\">10.4153/s0008414x25000021</a>","short":"S. Glock, J. Kim, L. Lichev, O. Pikhurko, S. Sun, Canadian Journal of Mathematics (2025) 1–43.","mla":"Glock, Stefan, et al. “On the (k + 2, k)-Problem of Brown, Erdős, and Sós for k = 5,6,7.” <i>Canadian Journal of Mathematics</i>, Cambridge University Press, 2025, pp. 1–43, doi:<a href=\"https://doi.org/10.4153/s0008414x25000021\">10.4153/s0008414x25000021</a>.","apa":"Glock, S., Kim, J., Lichev, L., Pikhurko, O., &#38; Sun, S. (2025). On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. <i>Canadian Journal of Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.4153/s0008414x25000021\">https://doi.org/10.4153/s0008414x25000021</a>","chicago":"Glock, Stefan, Jaehoon Kim, Lyuben Lichev, Oleg Pikhurko, and Shumin Sun. “On the (k + 2, k)-Problem of Brown, Erdős, and Sós for k = 5,6,7.” <i>Canadian Journal of Mathematics</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.4153/s0008414x25000021\">https://doi.org/10.4153/s0008414x25000021</a>."},"article_processing_charge":"No","OA_type":"hybrid","doi":"10.4153/s0008414x25000021","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"date_updated":"2025-09-30T10:28:07Z","status":"public","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"}}