{
  "abstract": "Let $m(n,p,q)$ be the largest size of a family of subsets of $[n]$ whose members pairwise intersect in at least $p$ vertices and omit at least $q$ common vertices. We prove that a product on disjoint sets attains this maximum whenever $p,q\\ge2$ and $n\\ge2^{15}(p+q-2)^3$. Together with the second author's result for $p=1$ or $q=1$, this establishes Frankl's Intersection--Union Product Conjecture in a wide range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25049",
  "author_total": 3,
  "authors": [
    {
      "name": "Fan Chang"
    },
    {
      "name": "Peter Frankl"
    },
    {
      "name": "Jian Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4vd9jpfsyyrp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25049",
  "fulltext_url": null,
  "id": "4vd9jpfsyyrp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25049",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223002767",
  "pdf_url": "https://arxiv.org/pdf/2610.25049",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4vd9jpfsyyrp3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25049",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25049"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proving the optimality of the product construction for the intersection--union problem",
  "title_text": "Proving the optimality of the product construction for the intersection--union problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/4vd9jpfsyyrp3",
  "venue": "arXiv",
  "versions": []
}