{
  "abstract": "Given an integer $k\\geq2$ and a graph $F$, the $k$-uniform suspension $\\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\\mathrm{ex}(n,\\mathcal{S}^kC_{2\\ell})$ and $\\mathrm{ex}(n,K^k_t,\\mathcal{S}^kC_{2\\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\\mathcal{S}^kC_{2\\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\\geq2$ and infinitely many $n$, \\[\\mathrm{ex}(n,K^{k}_{k+1},\\mathcal{S}^kC_4)=\\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\\mathrm{ex}(n,\\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\\geq2$ and $\\ell\\in\\{3,5\\}$, we determine the order of magnitude \\[\\mathrm{ex}(n,\\mathcal{S}^kC_{2\\ell})=Θ(n^{k-1+1/\\ell}).\\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\\mathrm{ex}(n,K^{k}_{k+1},\\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\\mathrm{ex}(n,\\mathcal{S}^kC_{2\\ell})$ with $\\ell\\in\\{3,5\\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22414",
  "author_total": 1,
  "authors": [
    {
      "name": "Dingyuan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1arbwb2k98x53.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22414",
  "fulltext_url": null,
  "id": "1arbwb2k98x53",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22414",
  "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": "W7204163537",
  "pdf_url": "https://arxiv.org/pdf/2608.22414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1arbwb2k98x53.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2608.22414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal problems for suspensions of even cycles",
  "title_text": "Extremal problems for suspensions of even cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/1arbwb2k98x53",
  "venue": "arXiv",
  "versions": []
}