{
  "abstract": "Two families $\\mathcal{F},\\mathcal{G}\\subseteq S_n$ are cross $t$-intersecting if every $σ\\in\\mathcal{F}$ and $τ\\in\\mathcal{G}$ agree on at least $t$ points. A $t$-coset is a coset of the stabilizer of $t$ points. A subset of $S_n$ is non-trivial if it is not contained in any $t$-coset. Let $d_m$ denote the $m$-th derangement number. We prove that, for all $t\\geq1$ and $n\\geq400t$, every pair of cross $t$-intersecting families $\\mathcal{F},\\mathcal{G}\\subseteq S_n$ satisfies the following: (i) $|\\mathcal{F}||\\mathcal{G}|\\leq((n-t)!-d_{n-t}-d_{n-t-1})((n-t)!+t)$ if $\\mathcal{F}\\cup\\mathcal{G}$ is non-trivial. (ii) $|\\mathcal{F}||\\mathcal{G}|\\leq((n-t)!-d_{n-t}-d_{n-t-1}+t)^2$ if both $\\mathcal{F}$ and $\\mathcal{G}$ are non-trivial. (iii) $\\min\\{|\\mathcal{F}\\setminus\\mathcal{C}|,|\\mathcal{G}\\setminus\\mathcal{C}|\\}\\leq t((n-t-1)!-(n-t-2)!)$ for some $t$-coset $\\mathcal{C}$ if $t\\geq2$. We also characterize all extremal configurations. The first result extends a theorem of Ellis (2011) to an exponentially wider range and sharpens the stability theorem of Keller, Lifshitz, Minzer and Sheinfeld (2024); the second gives a product version of the classical Hilton--Milner--Frankl theorem for permutations; and the third settles the remaining cases $t\\geq2$ of a conjecture of Ellis (2011) in a stronger form. In all three results, the linear dependence on $t$ is essentially optimal. Our proofs are based on the spread approximation method introduced by Kupavskii and Zakharov and on an approach to cross $t$-intersection problems developed by the present authors, with several essential refinements. As an application of our approach, we prove a product version of the Hilton--Milner--Frankl theorem for the alternating group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14409",
  "author_total": 2,
  "authors": [
    {
      "name": "Jie Wen"
    },
    {
      "name": "Benjian Lv"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7shdhe2bydbmt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14409",
  "fulltext_url": null,
  "id": "7shdhe2bydbmt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14409",
  "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": "W7213268699",
  "pdf_url": "https://arxiv.org/pdf/2609.14409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7shdhe2bydbmt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.14409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp stability for cross $t$-intersecting families of permutations in the linear range",
  "title_text": "Sharp stability for cross t-intersecting families of permutations in the linear range",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/7shdhe2bydbmt",
  "venue": "arXiv",
  "versions": []
}