{
  "abstract": "Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, for every pair of integers $r,k\\ge 3$, the number of $C_k^{(r)}$-free $r$-graphs on $n$ labelled vertices is \\[ 2^{(1+o(1))\\operatorname{ex}_{r}(n,C_{k}^{(r)})}. \\] While the analogous statement is known to fail for graphs ($r=2$), by a construction of Morris and Saxton, the general question remained open for hypergraphs. Very recently, Jiang and Longbrake answered the question affirmatively when $r\\geq5$. In this paper, we completely resolve the problem by proving that the answer is affirmative for every pair of integers $r,k\\geq3$. When $(r,k)\\neq(3,3)$, we establish balanced supersaturation results for linear cycles and combine them with the hypergraph container method. For $(r,k)=(3,3)$, we instead decompose $3$-graphs according to their pair-codegrees, encode the subhypergraph formed by the hyperedges that contain a large codegree pair as a directed graph, and apply a multicolour entropy theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38772",
  "author_total": 3,
  "authors": [
    {
      "name": "József Balogh"
    },
    {
      "name": "Ramon I. Garcia"
    },
    {
      "name": "Abhishek Methuku"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6xkrcz360pee3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38772",
  "fulltext_url": null,
  "id": "6xkrcz360pee3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38772",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215063547",
  "pdf_url": "https://arxiv.org/pdf/2609.38772",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6xkrcz360pee3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.38772",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38772"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting hypergraphs without linear cycles of fixed length",
  "title_text": "Counting hypergraphs without linear cycles of fixed length",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/6xkrcz360pee3",
  "venue": "arXiv",
  "versions": []
}