{
  "abstract": "We proved a log-star comparison between the expectation threshold $q(\\mathcal F)$ and the fractional expectation threshold $q_f(\\mathcal F)$ for any nontrivial increasing family $\\mathcal F$ on a finite ground set $V$ of size $|V|=N$. Specifically, we show that $$q_f(\\mathcal F)\\le64\\log_2^*(N+2)\\,q(\\mathcal F),$$ where $\\log_2^* x$ is the iterated logarithm of base-2. Using similiar methods, one can extend this result to the following stronger form: there exists a universal constant $C>0$ such that \\[ q_f(\\mathcal F)\\le Cq(\\mathcal F)\\log_2^*(l(\\mathcal F)),\\quad q_f(\\mathcal F)\\le Cq(\\mathcal F)\\log_2^*(\\operatorname{VC}(\\min\\mathcal F)), \\] where $l(\\mathcal F):=\\max\\{2,\\max_{S\\in\\min\\mathcal F}|S|\\}$ and $\\operatorname{VC}(\\min\\mathcal F)$ denotes the VC-dimension of $\\min\\mathcal F$. On October 7, 2026, OpenAI released a result proving the equivalence of fractional and integral thresholds, which supersedes the findings in this note. The authors are keeping this note for historical reference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30680",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuan Fang"
    },
    {
      "name": "Tianyu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ay6d8cwx073qj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30680",
  "fulltext_url": null,
  "id": "ay6d8cwx073qj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30680",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214628286",
  "pdf_url": "https://arxiv.org/pdf/2609.30680",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ay6d8cwx073qj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.30680",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30680"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold",
  "title_text": "A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/ay6d8cwx073qj",
  "venue": "arXiv",
  "versions": []
}