{
  "abstract": "Let $n>2k$ and $k\\ge4$. The largest size of an intersecting family of $k$-subsets of $[n]$ with covering number at least three is \\[ g_1(n,k)=\\binom{n-1}{k-1}-3\\binom{n-k-1}{k-1}+\\binom{n-k-2}{k-1}+\\binom{n-2k}{k-1}+3, \\] attained by a unique family up to isomorphism. We prove that every other maximal intersecting family $\\mathcal F$ with covering number three satisfies \\[ |\\mathcal F|\\le M(n,k)=\\max\\{q(n,k),g_2(n,k)\\}, \\] where \\begin{align*} q(n,k)&=\\binom{n-1}{k-1}-3\\binom{n-k-1}{k-1}+\\binom{n-k-3}{k-1}+\\binom{n-2k+1}{k-1}+3,\\\\ g_2(n,k)&=g_1(n,k)-\\binom{n-k-3}{k-3}+1. \\end{align*} This bound is attained for every $n>2k$ and $k\\ge4$, so $M(n,k)$ is the second largest distinct size. For $k=4$, $g_1(n,4)=13n-69$ and $M(n,4)=12n-61$. For $k\\ge5$, we have $M(n,k)=q(n,k)$ when $2k<n\\le3k-2$ and $M(n,k)=g_2(n,k)$ when $n\\ge3k-2$; the two terms coincide only at $n=3k-2$. We also classify all families attaining $M(n,k)$ that admit a critically shifted labeling, including additional nonisomorphic families at $n=2k+1$. The proof combines local replacements with an inverse-shift estimate and gives a new proof of the known maximum and its uniqueness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00149",
  "author_total": 1,
  "authors": [
    {
      "name": "Huajun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84cvrpw869bga.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00149",
  "fulltext_url": null,
  "id": "84cvrpw869bga",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00149",
  "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/2611.00149",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84cvrpw869bga.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00149",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00149"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Large Maximal Intersecting Families with Covering Number Three",
  "title_text": "Large Maximal Intersecting Families with Covering Number Three",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/84cvrpw869bga",
  "venue": "arXiv",
  "versions": []
}