{
  "abstract": "Let $π(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\\le k\\le r+1$. Sidorenko~(JCT-B, 2024) proved that $π(H_3^r)\\ge (1.7215-o(1))r^{-2}$ as $r\\to\\infty$ and $π(H_k^r)\\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\\to\\infty$. Clemen~later improved the first bound to $π(H_3^r)\\ge cr^{-2}\\sqrt{\\log r}$ for some constant $c>0$. In this article, we prove the following results. \\begin{itemize} \\item For any fixed $\\varepsilon>0$, there is a constant $c_\\varepsilon>0$ such that $$π(H_3^r)\\ge \\frac{c_\\varepsilon}{r(\\log r)^{2+\\varepsilon}}.$$ %$π(H_3^r)\\ge 1/(r(\\log r)^{2+o(1)})$. Together with the known upper bound $π(H_3^r)\\le1/r$, this implies $π(H_3^r)=r^{-1+o(1)}$. \\item For every $3\\le k\\le r+1$, let $s=\\min\\{k-2,r-k+2\\}$. Then \\begin{equation*} 0\\le \\frac{k-2}{r}-π(H_k^r) \\le \\frac{128}{r}\\left(\\sqrt{s\\log\\frac{er}{s}}+\\log\\frac{er}{s}\\right). \\end{equation*} This estimate yields several asymptotically sharp results for $π(H_k^r)$. For example, $π(H_k^r)=(1+o(1))(k-2)/r$ when $\\log(er/(k))=o(k)$. \\end{itemize}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18924",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiabao Yang"
    },
    {
      "name": "Xiutao Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3t46e7v2enme7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18924",
  "fulltext_url": null,
  "id": "3t46e7v2enme7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18924",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7203856598",
  "pdf_url": "https://arxiv.org/pdf/2608.18924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3t46e7v2enme7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2608.18924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Near-optimal Turán densities of $r$-graphs on $r+1$ vertices",
  "title_text": "Near-optimal Turán densities of r-graphs on r+1 vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/3t46e7v2enme7",
  "venue": "arXiv",
  "versions": []
}