{
  "abstract": "Let $g(n,k)$ be the maximum number of distinct sizes of maximal cliques in an $n$-vertex $k$-uniform hypergraph, and let $f(n,k)=n-g(n,k)$. We determine the asymptotic order of $f(n,k)$ for every fixed integer $k\\ge 3$. Define $L_2(x)=\\max\\{2,\\log_2(\\max\\{1,x\\})\\}$, and, for $j\\ge 3$, let $L_j(x)$ be the least number of iterations of $L_{j-1}$ needed to reach a value at most $16$. We prove that $$ f(n,k)=Θ_k(L_k(n)).$$ In particular, $f(n,3)=Θ(\\log^{*}n)$, where $\\log^{*}n$ denotes the iterated logarithm. We also determine the asymptotic behaviour of the layered-tree threshold $c(n,k)$ arising from Gao's insertion-tree method: $$ c(n,k)=\\log_2 L_k(n)+O_k(1). $$ Consequently, $f(n,k)=Θ_k\\!\\left(2^{c(n,k)}\\right)$. Our result gives a negative answer to Gao's question in the case $k=3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.27837",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiabao Yang"
    },
    {
      "name": "Lixiang Yu"
    },
    {
      "name": "Leilei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7awfejb9sr1vz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.27837",
  "fulltext_url": null,
  "id": "7awfejb9sr1vz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.27837",
  "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": "W7171990699",
  "pdf_url": "https://arxiv.org/pdf/2607.27837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7awfejb9sr1vz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2607.27837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.27837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the distinct maximal-clique sizes in $k$-uniform hypergraphs",
  "title_text": "On the distinct maximal-clique sizes in k-uniform hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/7awfejb9sr1vz",
  "venue": "arXiv",
  "versions": []
}