{
  "abstract": "The size of cross-intersecting families ${\\cal F}_1, \\dots, {\\cal F}_d$ is usually measured by the sum or the product of the sizes $|{\\cal F}_i|$. These two natural size functions have different extremal structures, and different techniques have been developed for them. We introduce the cross-degree of a size function and show that the potential extremal structures of maximum cross-intersecting families are the same for all size functions of finite cross-degree, including the sum and the product. Our main tool is a strengthened form of the generating family method of Ahlswede--Khachatrian, based on standard generating families and on local modifications of cross-intersecting families, which we call mutations. It treats two completely different types of extremal structures in a unified way and, unlike the original method, needs shifting in only one step.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38328",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziqing Xiang"
    },
    {
      "name": "Huajun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9x7m0nbkhx7yb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38328",
  "fulltext_url": null,
  "id": "9x7m0nbkhx7yb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38328",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38328",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9x7m0nbkhx7yb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38328",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38328"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sizes of cross-intersecting families",
  "title_text": "Sizes of cross-intersecting families",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/9x7m0nbkhx7yb",
  "venue": "arXiv",
  "versions": []
}