{
  "abstract": "The Brown--Erdős--Sós problem is a fundamental problem in sparse hypergraph Turán theory. For integers $r,k\\ge 2$ and $s\\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown, Erdős, and Sós proved that $f^{(r)}\\bigl(n;(r-t)k+t,k\\bigr)=Θ(n^t)$ for all $r>t\\ge 2$ and $k\\ge 2$. However, the existence and the value of the leading coefficient have remained largely open. We determine the coefficient $π(r,t,k):=\\lim_{n\\to\\infty}n^{-t}f^{(r)}\\bigl(n;(r-t)k+t,k\\bigr)$ for every such $r,t,k$, except when $(r,t)=(3,2)$ and $k\\ge 4$ is even. In particular, \\[ π(r,t,k) = \\begin{cases} \\frac{2}{t!\\bigl(2\\binom{r}{t}-1\\bigr)}, & \\text{if $k$ is odd},\\\\ \\frac{1}{t!\\binom{r}{t}}, & \\text{if $k$ is even and $r\\ge4$}. \\end{cases} \\] Surprisingly, for $r\\ge4$, the limit $π(r,t,k)$ depends on $k$ only through its parity, not its value. For the remaining case, we prove that $π(3,2,k)>1/6$ for all even $k\\ge 4$, showing that the natural packing construction is never asymptotically optimal. As an application, we extend a connection of Bennett, Cushman, and Dudek to arbitrary uniformity to resolve several cases of the Erdős--Gyárfás--Shelah generalized Ramsey problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38115",
  "author_total": 3,
  "authors": [
    {
      "name": "Ting-Wei Chao"
    },
    {
      "name": "Xinqi Huang"
    },
    {
      "name": "Hong Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5tc6s0rb3h00.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38115",
  "fulltext_url": null,
  "id": "c5tc6s0rb3h00",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38115",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214908573",
  "pdf_url": "https://arxiv.org/pdf/2609.38115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5tc6s0rb3h00.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:36Z",
      "native_id": "2609.38115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotics of the Brown--Erdős--Sós problem at integer exponents",
  "title_text": "Asymptotics of the Brown--Erdős--Sós problem at integer exponents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:36Z",
  "url": "https://research.draconitic.com/paper/c5tc6s0rb3h00",
  "venue": "arXiv",
  "versions": []
}