{
  "abstract": "A basis of a matroid is adapted to a flat if it contains a basis of the flat. We prove that a family of flats in a rank r matroid has a common adapted basis whenever every subfamily of at most r+1 flats does. The ground set and the family may be infinite; only the rank must be finite. This result, in particular, contains Klyachko's vector space Helly theorem for subspaces. As an application, we obtain a splitting criterion for tropical toric vector bundles on projective spaces. As a secondary application, we derive quadratic bounds for the Helly number for distributive generation in the lattice of flats.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36566",
  "author_total": 2,
  "authors": [
    {
      "name": "Amir Jafari"
    },
    {
      "name": "Kiumars Kaveh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exbxq8sv65eyd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36566",
  "fulltext_url": null,
  "id": "exbxq8sv65eyd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36566",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7224126046",
  "pdf_url": "https://arxiv.org/pdf/2610.36566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exbxq8sv65eyd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Helly theorems for lattice of flats of a matroid",
  "title_text": "Helly theorems for lattice of flats of a matroid",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/exbxq8sv65eyd",
  "venue": "arXiv",
  "versions": []
}