{
  "abstract": "We develop a new approach to cross-intersection problems in extremal set theory. The method builds on the iterative procedure introduced by Kupavskii and Zakharov (2024) and the $t$-cover method. It provides a flexible framework for deriving extremal and stability results for cross $t$-intersecting families. Our approach applies to a variety of combinatorial objects. As an application, we prove a product version of the seminal Erdős--Ko--Rado theorem for sufficiently spread set systems. Two families $\\mathcal{F}$ and $\\mathcal{G}$ of $k$-subsets of $[n]$ are called cross $t$-intersecting if $|F\\cap G|\\geq t$ for all $F\\in\\mathcal{F}$ and $G\\in\\mathcal{G}$. We determine the families maximizing $\\min\\{|\\mathcal{F}|, |\\mathcal{G}|\\}$ for large $n$ and all $t\\ge2$, generalizing results of Mörs (1985) and Füredi (1995) for cross $1$-intersecting families. We then determine the families maximizing $|\\mathcal{F}||\\mathcal{G}|$ under the condition $\\max\\{|\\cap_{F\\in\\mathcal{F}}F|,|\\cap_{G\\in\\mathcal{G}}G|\\}<t$ for large $n$. This improves the bound obtained by Frankl and Wang (2024), and provides a characterization of extremal configurations. For a family $\\mathcal{F}$ of subsets of $[n]$, we introduce its $t$-diversity $γ_t(\\mathcal{F})$, defined as the minimum number of sets from $\\mathcal{F}$ not containing a fixed $t$-subset. This serves as a natural generalization of the important notion of diversity for $t=1$. We obtain a stability result via $γ_t$, and determine the maximum of $\\min\\{γ_t(\\mathcal{F}),γ_t(\\mathcal{G})\\}$ for cross $t$-intersecting families $\\mathcal{F}$ and $\\mathcal{G}$. These yield new results for $t$-intersecting families, including a stability theorem towards a conjecture of Ellis, Keller and Lifshitz (2019), which may also be regarded as a $t$-intersection version, for large $n$, of an influential theorem of Frankl (1987).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.03315",
  "author_total": 2,
  "authors": [
    {
      "name": "Jie Wen"
    },
    {
      "name": "Benjian Lv"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/500tajm9bc7t9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.03315",
  "fulltext_url": null,
  "id": "500tajm9bc7t9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.03315",
  "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": "W7167614167",
  "pdf_url": "https://arxiv.org/pdf/2607.03315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/500tajm9bc7t9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.03315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.03315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability",
  "title_text": "A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/500tajm9bc7t9",
  "venue": "arXiv",
  "versions": []
}