{
  "abstract": "Using refined absorption, we prove that for every integer $g\\ge 1$ and real $γ> 0$, and for sufficiently large $n$, there exists an $n^{-1+γ}$-spread distribution on Latin squares of order $n$ and girth at least $g$ that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large $n$, there exists a subsquare-free Latin square of order $n$; indeed, it implies there exist at least $n^{(1-o(1))n^2}$ subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an $n^{-1+γ}$-spread distribution on high girth Latin squares.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28966",
  "author_total": 2,
  "authors": [
    {
      "name": "Thomas Lesgourgues"
    },
    {
      "name": "Luke Postle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g5kdpz89r04g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28966",
  "fulltext_url": null,
  "id": "7g5kdpz89r04g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28966",
  "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": "W7214342189",
  "pdf_url": "https://arxiv.org/pdf/2609.28966",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g5kdpz89r04g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28966",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28966"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Combinatorial Proof of Hilton's Conjecture and Beyond",
  "title_text": "A Combinatorial Proof of Hilton's Conjecture and Beyond",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7g5kdpz89r04g",
  "venue": "arXiv",
  "versions": []
}