{
  "abstract": "We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.",
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  "article_type": null,
  "arxiv": "2610.02833",
  "author_total": 3,
  "authors": [
    {
      "name": "Deep Bhattacharjee"
    },
    {
      "name": "Priyabrata Mandal"
    },
    {
      "name": "Ushashi Bhattacharya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9red78a4at97e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02833",
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  "id": "9red78a4at97e",
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  "landing_url": "https://arxiv.org/abs/2610.02833",
  "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"
    ],
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      "arxiv",
      "math.NT"
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      "openalex",
      "2607|Limits and Structures in Graph Theory"
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  "openalex": "W7220430360",
  "pdf_url": "https://arxiv.org/pdf/2610.02833",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9red78a4at97e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02833",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02833"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures",
  "title_text": "Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/9red78a4at97e",
  "venue": "arXiv",
  "versions": []
}