{
  "abstract": "Given a graph $F$, a family $\\mathcal F$ of graphs on $[n]$ is \\emph{$F$-intersecting} if $G\\cap H$ contains a copy of $F$ for every $G,H\\in\\mathcal F$. We prove that there exists an absolute constant $\\varepsilon>0$ such that every $P_4$-intersecting family $\\mathcal F$ satisfies $|\\mathcal F|\\le\\left(\\frac12-\\varepsilon\\right)2^{\\binom n2}$, which resolves a conjecture of Alon. Combined with Alon's reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1/2$ if and only if $F$ is a star forest. Multi-colored versions are also considered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13145",
  "author_total": 2,
  "authors": [
    {
      "name": "Jie Han"
    },
    {
      "name": "Bin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/491g652bg0g42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13145",
  "fulltext_url": null,
  "id": "491g652bg0g42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13145",
  "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": "W7212544057",
  "pdf_url": "https://arxiv.org/pdf/2609.13145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/491g652bg0g42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.13145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Alon's conjecture on intersecting families of graphs",
  "title_text": "Alon's conjecture on intersecting families of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/491g652bg0g42",
  "venue": "arXiv",
  "versions": []
}