{
  "abstract": "The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\\ge2$ fixed and $n\\to\\infty$; then it is easy to show that the extremal function grows quadratically in $n$. Delcourt and Postle proved that the limit $π(k):=\\lim_{n\\to\\infty} f^{(3)}(n;k+2,k)/n^2$ exists for every $k$. While Brown, Erdős and Sós observed that $π(2)=1/6$ already in the 1970s, the value of $π(k)$ for $3\\le k\\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $π(k)$ for every odd $k$. Independently of the last result, we show that $π(9)=1/5$. Also, we prove that $π(8)\\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $π(8)\\ge 3/16$. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on $π(9)$ is based on a finite field construction combined with existing packing results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03323",
  "author_total": 5,
  "authors": [
    {
      "name": "Levente Bodnár"
    },
    {
      "name": "Oleg Pikhurko"
    },
    {
      "name": "Shumin Sun"
    },
    {
      "name": "Yan Wang"
    },
    {
      "name": "Jiasheng Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b21n3161x7j37.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03323",
  "fulltext_url": null,
  "id": "b21n3161x7j37",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03323",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.03323",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b21n3161x7j37.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges",
  "title_text": "The quadratic Brown--Erdős--Sós problem for 3-uniform hypergraphs with 8 and 9 edges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/b21n3161x7j37",
  "venue": "arXiv",
  "versions": []
}