{
  "abstract": "Let $\\TP_k$ denote the triangular pyramid with $k$ layers. For a graph family $\\mathcal F$, let $\\ex(n,\\mathcal F)$ be the maximum number of edges in an $n$-vertex graph containing no member of $\\mathcal F$ as a subgraph. For a graph $H$, let $\\cJ(H)$ be the family of subgraph-minimal graphs among all such graphs $H-I$, where $I\\subseteq V(H)$ is an independent set. For every integer $k\\ge4$ and all sufficiently large $n$, we prove that \\[ \\ex(n,\\TP_k)=\\max_{0\\le m\\le n}\\{m(n-m)+\\ex(m,\\cJ(\\TP_k))\\}+O_k(n)=\\left\\lfloor n^2/4\\right\\rfloor+Θ_k(n^{4/3}). \\] Moreover, for $k=4$, we characterize all extremal graphs and show that \\[ \\ex(n,\\TP_4)=\\max_{0\\le m\\le n}\\bigl\\{m(n-m)+\\ex\\big(m,\\cJ(\\TP_4)\\big)\\bigr\\} \\] for sufficiently large $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18493",
  "author_total": 3,
  "authors": [
    {
      "name": "Chuanqi Xiao"
    },
    {
      "name": "Long-Tu Yuan"
    },
    {
      "name": "Xiaowen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0jg6bne34918q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18493",
  "fulltext_url": null,
  "id": "0jg6bne34918q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18493",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221770500",
  "pdf_url": "https://arxiv.org/pdf/2610.18493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0jg6bne34918q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Turán numbers of triangular pyramids",
  "title_text": "Turán numbers of triangular pyramids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/0jg6bne34918q",
  "venue": "arXiv",
  "versions": []
}