---
OA_place: publisher
OA_type: hybrid
_id: '19017'
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Stefan
  full_name: Glock, Stefan
  last_name: Glock
- first_name: Jaehoon
  full_name: Kim, Jaehoon
  last_name: Kim
- first_name: Lyuben
  full_name: Lichev, Lyuben
  id: 9aa8388e-d003-11ee-8458-c4c1d7447977
  last_name: Lichev
- first_name: Oleg
  full_name: Pikhurko, Oleg
  last_name: Pikhurko
- first_name: Shumin
  full_name: Sun, Shumin
  last_name: Sun
citation:
  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>
  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>.
  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.
  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.
  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>.
  short: S. Glock, J. Kim, L. Lichev, O. Pikhurko, S. Sun, Canadian Journal of Mathematics
    (2025) 1–43.
date_created: 2025-02-10T08:39:46Z
date_published: 2025-01-06T00:00:00Z
date_updated: 2025-09-30T10:28:07Z
day: '06'
ddc:
- '500'
department:
- _id: MaKw
doi: 10.4153/s0008414x25000021
external_id:
  arxiv:
  - '2403.04474'
  isi:
  - '001416788600001'
has_accepted_license: '1'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.4153/s0008414x25000021
month: '01'
oa: 1
oa_version: Published Version
page: 1-43
publication: Canadian Journal of Mathematics
publication_identifier:
  eissn:
  - 1496-4279
  issn:
  - 0008-414X
publication_status: epub_ahead
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7
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)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
year: '2025'
...