{
  "abstract": "Bradač, Sudakov, and Wigderson characterized $p$-locally dense graphons by a quadratic inequality for all bounded nonnegative functions. Their proof uses Reiher's finite lemma and graphon approximation, and they asked for a direct proof. We give one by rounding simple functions. Divide each level set into $m$ equal-measure pieces and retain each piece independently with probability equal to its level. Off-diagonal terms agree in expectation; the diagonal error is at most $\\norminf{W-p}/(4m)$. Letting $m\\to\\infty$ and then approximating in $L^1$ proves the inequality. A three-atom example shows that atomlessness is necessary if arbitrary probability spaces are allowed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19014",
  "author_total": 1,
  "authors": [
    {
      "name": "Dean Menezes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d72bbcn563201.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19014",
  "fulltext_url": null,
  "id": "d72bbcn563201",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19014",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213503879",
  "pdf_url": "https://arxiv.org/pdf/2609.19014",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d72bbcn563201.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19014",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19014"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Direct Proof of the Locally Dense Graphon Inequality",
  "title_text": "A Direct Proof of the Locally Dense Graphon Inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/d72bbcn563201",
  "venue": "arXiv",
  "versions": []
}