{
  "abstract": "For $H$ a hypergraph, the Hall ratio $ρ(H)=\\text{max}_{X\\subseteq V(H)}\\frac{\\lvert{X}\\rvert}{α(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromatic number $2$. Nešetřil, Rödl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k \\geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20562",
  "author_total": 4,
  "authors": [
    {
      "name": "Veronica Bitonti"
    },
    {
      "name": "Freddie Illingworth"
    },
    {
      "name": "Pedram Noohi"
    },
    {
      "name": "Jane Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eya50c6s3zqyv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20562",
  "fulltext_url": null,
  "id": "eya50c6s3zqyv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20562",
  "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": "W7222249009",
  "pdf_url": "https://arxiv.org/pdf/2610.20562",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eya50c6s3zqyv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20562",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20562"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hypergraphs with the smallest Hall ratio are still hard to colour",
  "title_text": "Hypergraphs with the smallest Hall ratio are still hard to colour",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/eya50c6s3zqyv",
  "venue": "arXiv",
  "versions": []
}