{
  "abstract": "We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08291",
  "author_total": 1,
  "authors": [
    {
      "name": "Yunjiang Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59yybnwfaeqzg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08291",
  "fulltext_url": null,
  "id": "59yybnwfaeqzg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08291",
  "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": "W7212064945",
  "pdf_url": "https://arxiv.org/pdf/2609.08291",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59yybnwfaeqzg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08291",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08291"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Entropy bounds and global couplings for union-closed families",
  "title_text": "Entropy bounds and global couplings for union-closed families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/59yybnwfaeqzg",
  "venue": "arXiv",
  "versions": []
}