{
  "abstract": "We study the structure of large $t$-intersecting families. A family of $k$-subsets of an $n$-set is $t$-intersecting if every two of its members intersect in at least $t$ elements. A $t$-intersecting family is non-trivial if no $t$-subset is contained in all its members. We prove several stability results for the seminal Hilton--Milner--Frankl theorem. First, for any fixed $η,\\varepsilon,θ\\in(0,1)$, we prove that if $k/t\\geq1+η$ and $n=Ω(tk^{1+\\varepsilon})$, then every non-trivial $t$-intersecting family of size greater than $(1+θ)|\\mathcal{K}|$ is a subfamily of one of the two extremal families in the theorem, where $\\mathcal{K}$ is an explicit large non-trivial $t$-intersecting family. The key ingredient in the proof is a removal lemma. We also obtain a classification of all $t$-intersecting families with size bounded below by $|\\mathcal{K}|$ minus an explicit lower-order term, provided that $k\\geq t+4\\geq6$ and $n\\geq t+6\\cdot\\max\\{(t+2)^2, k(k-t)\\}$. This strengthens results of Cao--Lv--Wang (2021) and Frankl (2025) for a broad range of $k$ and $t$ (for example, when $k-t\\geq2\\sqrt{t}$). As an application of this classification, we determine the largest $t$-intersecting families for each prescribed lower bound on $t$-diversity not exceeding $t(n-k)$, thereby obtaining $t$-intersection versions of results of Han and Kohayakawa (2017) and Kupavskii (2025). To establish these results, we develop techniques based on the spread approximation method and the $t$-cover method, which may be useful for other intersection problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14197",
  "author_total": 2,
  "authors": [
    {
      "name": "Jie Wen"
    },
    {
      "name": "Benjian Lv"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aryd8dph4chtg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14197",
  "fulltext_url": null,
  "id": "aryd8dph4chtg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14197",
  "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": "W7203635008",
  "pdf_url": "https://arxiv.org/pdf/2608.14197",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aryd8dph4chtg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.14197",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14197"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Structure of large $t$-intersecting families I: Stability for the Hilton--Milner--Frankl theorem",
  "title_text": "Structure of large t-intersecting families I: Stability for the Hilton--Milner--Frankl theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/aryd8dph4chtg",
  "venue": "arXiv",
  "versions": []
}