{
  "abstract": "Let $n\\geq2\\ell\\geq2k\\geq6$ and $n>k+\\ell$. We determine the maximum of $|\\mathcal A||\\mathcal B|$ over all nonempty, nontrivial cross-intersecting families $\\mathcal A\\subseteq\\binom{[n]}k$ and $\\mathcal B\\subseteq\\binom{[n]}\\ell$. The exact bound is $|\\mathcal A||\\mathcal B|\\leq\\max\\{M_1,M_2\\}$, where \\[ \\begin{aligned} M_1 &=\\left(\\binom{n-1}{k-1}-\\binom{n-\\ell-1}{k-1}+1\\right) \\left(\\binom{n-1}{\\ell-1}-\\binom{n-k-1}{\\ell-1}+1\\right),\\\\ M_2 &=\\left(\\binom{n-2}{\\ell-2}+1\\right) \\sum_{i=1}^{2} \\left(\\binom{n-i}{k-1}-\\binom{n-\\ell-i}{k-1}\\right). \\end{aligned} \\] Both terms are attained by explicit constructions: a Hilton--Milner-type pair and a two-point construction, respectively. We classify all equality cases, including the additional cross-triangle pairs when $k=\\ell=3$. Two structural results underlie the proof. First, we establish a restricted-lex comparison theorem showing that restricted lexicographic initial segments minimize disjointness neighborhoods, and characterize equality at the first nontrivial threshold. Second, a decomposition of critically shifted families into two parts with disjoint cores leads to a sharp product theorem, with equality characterized, for all cross-intersecting pairs admitting this decomposition, without additional nontriviality assumptions. Endpoint comparisons and reverse shifting complete the proof for unequal as well as equal uniformities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34944",
  "author_total": 2,
  "authors": [
    {
      "name": "Biao Wu"
    },
    {
      "name": "Huajun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25gja06j0h95h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34944",
  "fulltext_url": null,
  "id": "25gja06j0h95h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34944",
  "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": "W7223985032",
  "pdf_url": "https://arxiv.org/pdf/2610.34944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25gja06j0h95h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34944"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the product of nontrivial cross-intersecting families",
  "title_text": "On the product of nontrivial cross-intersecting families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/25gja06j0h95h",
  "venue": "arXiv",
  "versions": []
}