{
  "abstract": "Uniform Turán density problems ask for the minimum density threshold that guarantees a copy of a fixed 3-uniform hypergraph $F$ in every sufficiently large 3-uniform hypergraph whose edges are uniformly distributed. This line of research was initiated by Erdős and Sós in the 1980s, who introduced the notion of \\emph{vertex-set Turán density} $π_{vvv}(F)$. Reiher, Rödl and Schacht later defined two related concepts, namely, the \\emph{vertex-pair Turán density} $π_{ve}(F)$ and the \\emph{cherry Turán density} $π_{ee}(F)$, which satisfy the chain of inequalities $π_{vvv}(F)\\geπ_{ve}(F)\\geπ_{ee}(F)$. In this paper, we investigate the problem for their vanishing densities. First, we characterize the 3-uniform hypergraphs $F$ for which $π_{ee}(F)=0$, thereby answering a question posed by Schacht. We show that these are precisely the so-called \\emph{max-layered} 3-uniform hypergraphs recently introduced by Lamaison. Furthermore, we answer an analogous question of Lamaison for the \\emph{minimum-codegree palette density} $π_{ee}^{\\rm pal}$. Together, these results imply that the following are equivalent: $π_{ee}(F)=0$, $π_{ee}^{\\rm pal}(F)=0$, and $F$ is a max-layered 3-uniform hypergraph. Let $\\star \\in \\{vvv, ve, ee \\}$. We say that zero is a \\emph{jump} for $π_{\\star}$ if there exists $c>0$ such that no $3$-uniform hypergraph $F$ satisfies $π_{\\star}(F)\\in(0,c)$. A famous result of Reiher, Rödl and Schacht asserts that zero is a jump for $π_{vvv}$. In contrast, we prove that zero is not a jump for either $π_{ve}$ or $π_{ee}$. This also yields a negative answer to Schacht's well-ordering question for $π_{ve}$. Together with existing results, these findings complete the picture for all three notions of uniform Turán density at zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28574",
  "author_total": 3,
  "authors": [
    {
      "name": "Hao Lin"
    },
    {
      "name": "Jie Ma"
    },
    {
      "name": "Wenling Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30f6zxybvvbc1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28574",
  "fulltext_url": null,
  "id": "30f6zxybvvbc1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28574",
  "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": "W7223123568",
  "pdf_url": "https://arxiv.org/pdf/2610.28574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30f6zxybvvbc1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Vanishing Uniform Turán Densities: Characterization and Jumps",
  "title_text": "Vanishing Uniform Turán Densities: Characterization and Jumps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/30f6zxybvvbc1",
  "venue": "arXiv",
  "versions": []
}