{
  "abstract": "Improving an earlier bound of Frieze and Mubayi, Iliopoulos showed that any $k$-uniform uncrowded hypergraph of maximum degree $Δ$ has list chromatic number at most $(1+o(1))(k-1)\\Big(\\fracΔ{\\logΔ}\\Big)^{\\frac{1}{k-1}}$. Determining the optimal leading constant in this bound remains a major open problem. A natural target for the constant, suggested by the coupon-collector heuristic, is $(k-1)^{\\frac{1}{k-1}}$, which also arises independently as the shattering threshold for coloring and finding large independent sets in sparse random hypergraphs. We show that uncrowded hypergraphs attain this threshold: for $k \\ge 2$, every $k$-uniform uncrowded hypergraph $\\mathcal{H}$ of maximum degree at most $Δ$ satisfies $$χ_\\ell(\\mathcal{H})\\le(1+o(1))\\Big((k-1)\\fracΔ{\\logΔ}\\Big)^{\\frac{1}{k-1}}.$$ This matches the coupon-collector heuristic, and establishes a 2019 conjecture of Molloy for uncrowded hypergraphs. As a consequence, we show that every $k$-uniform linear hypergraph of maximum degree at most $Δ$ has chromatic number at most $c_k\\Big(\\fracΔ{\\log Δ}\\Big)^{\\frac{1}{k-1}}$ with $c_k=1-o_k(1)$, resolving, in a stronger form, a recent conjecture of Verstraëte and Wilson on the independence number of linear hypergraphs. Our techniques yield the first palette sparsification result for hypergraph coloring using $o(Δ^{\\frac{1}{k-1}})$ colors per vertex. Specifically, for an uncrowded $k$-graph $\\mathcal{H}$ and $q=Θ\\Big({\\Big(\\fracΔ{γ\\logΔ}\\Big)^{\\frac{1}{k-1}}}\\Big)$, sampling $Θ(Δ^{\\fracγ{k-1}})$ colors per vertex suffices to obtain a proper $q$-coloring of $\\mathcal{H}$ w.h.p. As an algorithmic consequence, we obtain a single-pass streaming algorithm using $O\\big(n^{1+o(1)}\\big)$ space for uncrowded $k$-graphs, and, via hypergraph partitioning, for linear hypergraphs as well.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.18048",
  "author_total": 3,
  "authors": [
    {
      "name": "Abhishek Dhawan"
    },
    {
      "name": "Abhishek Methuku"
    },
    {
      "name": "Minh-Quan Vo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1f7yzgzbx951k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.18048",
  "fulltext_url": null,
  "id": "1f7yzgzbx951k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.18048",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7164991767",
  "pdf_url": "https://arxiv.org/pdf/2606.18048",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1f7yzgzbx951k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.18048",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.18048"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "List coloring uncrowded hypergraphs at the shattering threshold",
  "title_text": "List coloring uncrowded hypergraphs at the shattering threshold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1f7yzgzbx951k",
  "venue": "arXiv",
  "versions": []
}