{
  "abstract": "Kalai conjectured that the largest intersecting family of triangulations of an $n$-gon is the family consisting of all triangulations that contain a fixed ear (a chord of length $2$). What happens if we strengthen the intersecting condition to the condition that any two of the triangulations must intersect on a chord from a specified family of chords -- for example, what if any two triangulations intersect on a chord from some fixed point of the $n$-gon, or intersect on an ear? In this paper we verify Kalai's conjecture for families of triangulations in which any two intersect on a chord from some fixed point. We also show that any ear-intersecting family of triangulations has size at most $T_{n-1} +T_{n-3} = T_{n-1}(17/16 + o(1))$, where $T_{n-1}$ is the upper bound in Kalai's conjecture, namely the number of triangulations of an $(n-1)$-gon. This is based on an exact upper bound of $T_{n-1}$ for the size of a family of triangulations in which any two intersect on some ear from the set of all ears with one ear excluded.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39116",
  "author_total": 2,
  "authors": [
    {
      "name": "Imre Leader"
    },
    {
      "name": "Ta Sheng Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a7tm3kfrfbf8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39116",
  "fulltext_url": null,
  "id": "2a7tm3kfrfbf8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39116",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39116",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a7tm3kfrfbf8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39116",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39116"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Some results on intersecting families of triangulations",
  "title_text": "Some results on intersecting families of triangulations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2a7tm3kfrfbf8",
  "venue": "arXiv",
  "versions": []
}