{
  "abstract": "The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\\mathrm{UC}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\\mathrm L}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound $c_{\\mathrm{UC}}\\ge c_{\\mathrm L}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\\mathrm{UC}}\\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\\ldots$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02295",
  "author_total": 2,
  "authors": [
    {
      "name": "Simone Costa"
    },
    {
      "name": "Ankan Sadhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ds39vr03vwr9s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02295",
  "fulltext_url": null,
  "id": "ds39vr03vwr9s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02295",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220451236",
  "pdf_url": "https://arxiv.org/pdf/2610.02295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ds39vr03vwr9s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Beyond Liu's 0.382709 threshold for the union-closed sets conjecture",
  "title_text": "Beyond Liu's 0.382709 threshold for the union-closed sets conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/ds39vr03vwr9s",
  "venue": "arXiv",
  "versions": []
}