{
  "abstract": "In 1966, Tutte defined $k$-connected matroids for each $k \\ge 2$. Robertson and Seymour built on Tutte's idea to define tangles as a way to identify highly connected parts of a matroid. Motivated by tangles, a set $Z$ in a matroid $M$ has been defined to be $3$-connected if, for each $j$ in $\\{1,2\\}$, there is no $j$-separation $(X,Y)$ of $M$ such that each of $|Z \\cap X|$ and $|Z \\cap Y|$ is at least $j$. This paper solves a problem of Whittle for $k= 3$ by proving that a minor-minimal matroid having $Z$ as a $3$-connected set has at most $3|Z| - 4$ elements. We also identify the matroids for which this bound is attained.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29979",
  "author_total": 2,
  "authors": [
    {
      "name": "Sean Boros"
    },
    {
      "name": "James Oxley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11zn7tahvtgnm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29979",
  "fulltext_url": null,
  "id": "11zn7tahvtgnm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29979",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223202517",
  "pdf_url": "https://arxiv.org/pdf/2610.29979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11zn7tahvtgnm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29979"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On $k$-connected Sets in Matroids",
  "title_text": "On k-connected Sets in Matroids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/11zn7tahvtgnm",
  "venue": "arXiv",
  "versions": []
}