{
  "abstract": "We show that the uniform probability measure on copies of a nonempty graph $H$ in $K_n$ is $Cq_H\\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold. Consequently, the fractional expectation threshold of $H$ is at most $Cq_H\\log(2e(H))$. We remove the logarithmic factor for trees and for graphs whose average degree is at least the logarithm of their maximum degree. This proves the ``second'' Kahn-Kalai conjecture for these two classes, which encompass most of the standard families studied in random graph containment problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20546",
  "author_total": 1,
  "authors": [
    {
      "name": "Tuan Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2sb7e6zc5m5hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20546",
  "fulltext_url": null,
  "id": "2sb7e6zc5m5hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20546",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
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  ],
  "openalex": "W7213580361",
  "pdf_url": "https://arxiv.org/pdf/2609.20546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2sb7e6zc5m5hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.20546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Fractional expectation thresholds and the \"second\" Kahn-Kalai conjecture",
  "title_text": "Fractional expectation thresholds and the \"second\" Kahn-Kalai conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2sb7e6zc5m5hy",
  "venue": "arXiv",
  "versions": []
}