{
  "abstract": "We prove Chvátal's conjecture, posed in 1972: every hereditary family of subsets of a finite set has a largest intersecting subfamily that is a star. More generally, we prove a sharp correlation inequality for increasing Boolean functions $f,g:\\{0,1\\}^n\\to\\{0,1\\}$. Writing $g^*(x)=1-g(1-x)$, we show that $$ \\sum_{\\varnothing\\ne S\\subseteq[n]}\\hat{g}(S)^2\\max_{i\\in S}\\mathrm{Inf}_i[f]\\le\\frac{2\\mathrm{Cov}(f,g)\\mathrm{Cov}(f,g^*)}{\\mathrm{Cov}(f,g)+\\mathrm{Cov}(f,g^*)}. $$ When $g$ is antipodal, that is, $g=g^*$, this yields $\\mathrm{Cov}(f,g)\\ge\\frac{1}{4}\\min_{i\\in[n]}\\mathrm{Inf}_i[f]$, the correlation formulation of Chvátal's conjecture due to Friedgut, Kahn, Kalai and Keller.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19123",
  "author_total": 3,
  "authors": [
    {
      "name": "Fan Chang"
    },
    {
      "name": "Hong Liu"
    },
    {
      "name": "Miao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2w38as5s1m94s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19123",
  "fulltext_url": null,
  "id": "2w38as5s1m94s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19123",
  "language": null,
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  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.FA"
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      "arxiv",
      "math.PR"
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      "openalex",
      "2607|Advanced Combinatorial Mathematics"
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  "openalex": "W7213512071",
  "pdf_url": "https://arxiv.org/pdf/2609.19123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2w38as5s1m94s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A proof of Chvátal's conjecture via a sharp correlation inequality",
  "title_text": "A proof of Chvátal's conjecture via a sharp correlation inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/2w38as5s1m94s",
  "venue": "arXiv",
  "versions": []
}