{
  "abstract": "Montgomery recently proved that for sufficiently large $n$, every Latin square of order $n$ has a partial transversal with $n-1$ cells, and every Steiner triple system of order $n$ has a matching with $\\lfloor n/3\\rfloor-1$ edges, thus confirming the Ryser--Brualdi--Stein conjecture for even $n$ and the conjecture of Brouwer. We prove sharp enumerative refinements of these results: there is an absolute constant $c>0$ such that, for sufficiently large $n$, 1) every Latin square of order $n$ has $ \\left((1\\pm n^{-c})\\frac{n}{\\mathrm {e}^2}\\right)^n$ partial transversals with $n-1$ cells; 2) every Steiner triple system of order $n$ has $ \\left((1\\pm n^{-c})\\frac{n}{2\\mathrm {e}^2}\\right)^{\\lfloor n/3\\rfloor}$ matchings with $\\lfloor n/3\\rfloor-1$ edges. The first estimate confirms predictions of Montgomery and Kelly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11006",
  "author_total": 2,
  "authors": [
    {
      "name": "Yantao Tang"
    },
    {
      "name": "Yi Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6f61v5makgcs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11006",
  "fulltext_url": null,
  "id": "a6f61v5makgcs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11006",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212272424",
  "pdf_url": "https://arxiv.org/pdf/2609.11006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6f61v5makgcs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems",
  "title_text": "Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/a6f61v5makgcs",
  "venue": "arXiv",
  "versions": []
}