{
  "abstract": "For every fixed $k\\ge 3$ and every integer $b\\ge 2$, we construct a finite vertex-partitioned $k$-uniform hypergraph with blocks of size $b$ and no independent transversal. We show that the maximum number of edges meeting a block is $\\left(\\frac{(k-1)^{k-1}}{k^k}+O_k(b^{-1})\\right)b^k,$ while the maximum vertex degree is $\\left(\\frac{(k-2)^{k-2}}{(k-1)^{k-1}}+O_k(b^{-1})\\right)b^{k-1}$. This shows that the known bound on the number of edges meeting a block is asymptotically sharp even when the leading constant in the maximum vertex degree tends to zero, which answers a problem of Groenland, Kaiser, Treffers, and Wales. By blowing up the same construction, we further obtain for arbitrarily large $n$, there exists a $k$-uniform hypergraph with $n$ blocks, each of size at least $n$, and $Δ(G)<\\left(1-\\frac1k\\right)^{k-1}n^{k-1}$, but with no independent transversal. This gives a negative answer to a problem of Haxell and Wdowinski.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22993",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinzhe Song"
    },
    {
      "name": "Xia Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pebrnq68ja0v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22993",
  "fulltext_url": null,
  "id": "0pebrnq68ja0v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22993",
  "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": "W7222935172",
  "pdf_url": "https://arxiv.org/pdf/2610.22993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pebrnq68ja0v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hypergraphs of Low Degree Without Independent Transversals",
  "title_text": "Hypergraphs of Low Degree Without Independent Transversals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/0pebrnq68ja0v",
  "venue": "arXiv",
  "versions": []
}