{
  "abstract": "A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\\to\\infty$. For $n=2k+2$, we prove that almost all intersecting families are trivial and that their number is \\[ (2k+2+o(1))\\,2^{\\binom{2k+1}{k-1}}. \\] Together with Yang's recent result for $n\\ge 2k+3$, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For $n=2k+1$, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in $k-1$ elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02119",
  "author_total": 1,
  "authors": [
    {
      "name": "Lina Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ejx7btv2zvj6w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02119",
  "fulltext_url": null,
  "id": "ejx7btv2zvj6w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02119",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215992960",
  "pdf_url": "https://arxiv.org/pdf/2610.02119",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ejx7btv2zvj6w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.02119",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02119"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Typical intersecting families at $n=2k+1$ and $n=2k+2$",
  "title_text": "Typical intersecting families at n=2k+1 and n=2k+2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/ejx7btv2zvj6w",
  "venue": "arXiv",
  "versions": []
}