{
  "abstract": "Let $p_c(\\mathcal F)$, $q(\\mathcal F)$, and $q_f(\\mathcal F)$ denote the threshold, expectation threshold, and fractional expectation threshold of a family $\\mathcal F$ of nonempty subsets of a finite set, respectively. We prove that there is an absolute constant $C>0$ such that, if $\\mathcal F$ has VC dimension at most $d\\ge1$, then $p_c(\\mathcal F)\\le Cq(\\mathcal F)\\log(d+1)$. More generally, for every $0<\\varepsilon\\le1/2$, a binomial random set of density $\\min\\{1,Cq(\\mathcal F)\\log((d+1)/\\varepsilon)\\}$ contains a member of $\\mathcal F$ with probability at least $1-\\varepsilon$. Consequently, $q_f(\\mathcal F)\\le Cq(\\mathcal F)\\log(d+1)$, verifying Talagrand's integral--fractional conjecture for families of any fixed VC dimension. We also prove that if a $k$-spread probability measure has support of VC dimension at most $d$, then a binomial random set of density $\\min\\{1,(C/k)\\log((d+1)/\\varepsilon)\\}$ contains a member of its support with probability at least $1-\\varepsilon$. In both random-containment results, the factor \\(\\log((d+1)/\\varepsilon)\\) is optimal up to absolute constants. As an application of the spread theorem, we prove that every $n$-uniform family of VC dimension at most $d$ with more than $(C p^{-1}\\log((d+1)/\\varepsilon))^n$ members contains a $(p,\\varepsilon)$-robust sunflower. In particular, every such family with more than $(Cr\\log(d+1))^n$ members contains an $r$-sunflower, improving the recent bound $(Crd)^n$ of Ge, Wang, Xu, and Zhao.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30263",
  "author_total": 1,
  "authors": [
    {
      "name": "Chong Shangguan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90gqj05ejvqzc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30263",
  "fulltext_url": null,
  "id": "90gqj05ejvqzc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30263",
  "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": "W7214364543",
  "pdf_url": "https://arxiv.org/pdf/2609.30263",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90gqj05ejvqzc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30263",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30263"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Thresholds and spread in set systems of bounded VC-dimension",
  "title_text": "Thresholds and spread in set systems of bounded VC-dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/90gqj05ejvqzc",
  "venue": "arXiv",
  "versions": []
}