{
  "abstract": "Let $\\Palt=(\\C,\\T)$ be a $k$-palette. For $0\\le t\\le k-1$, its $t$th coordinate-extension degree is the minimum, over every choice of $t$ coordinates and every assignment of colors to them, of the proportion of assignments to the remaining $k-t$ coordinates that complete the fixed colors to an admissible $k$-tuple. For a $k$-graph $F$, we define $π_t^{\\ext}(F)$ as the supremum of this degree over all palettes not admitted by $F$. We prove that \\[ π_t^{\\ext}(F)=0 \\quad\\text{if and only if}\\quad F\\text{ is }t\\text{-layered}. \\] We also relate $t$-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At $t=0$, the parameter $π_0^{\\ext}(F)$ is the $(k-2)$-uniform Turán density $π_{k-2}(F)$. For every $k\\ge3$ and $r\\ge2$, we construct a finite $k$-graph $F_{k,r}$ with \\( π_{k-2}(F_{k,r})=2(r-1)/rk^k. \\) Thus $2/k^k$ is an accumulation point for single forbidden $k$-graphs. We also show that the least density of a $k$-graph that fails condition $\\Sp$ of Lin, Wang and Zhou is $4/(3k^k)$. Finally, for every admissible matching of size $m$, we construct a $k$-graph that satisfies $\\Sp$ for every coordinate pair, has no vanishing order, and has density $2^m/k^k$. This disproves a conjecture of Lin, Wang and Zhou for every $k\\ge3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03347",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiabao Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/65hw8tqmz363t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03347",
  "fulltext_url": null,
  "id": "65hw8tqmz363t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03347",
  "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": "W7220448979",
  "pdf_url": "https://arxiv.org/pdf/2610.03347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/65hw8tqmz363t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Coordinate-extension degrees and layered $k$-uniform hypergraphs",
  "title_text": "Coordinate-extension degrees and layered k-uniform hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/65hw8tqmz363t",
  "venue": "arXiv",
  "versions": []
}