{
  "abstract": "For every odd integer $r\\ge1$, let $q_r=1-2^{-r}\\binom r{(r-1)/2}$. We show that, for every odd $r\\ge3$ and sufficiently large $n$, every $n$-vertex $r$-uniform hypergraph with at least $q_r\\binom nr$ edges contains a tight component with at least $(1/2-o(1))\\binom nr$ edges. This confirms a conjecture posed by Lang and Sanhueza-Matamala in 2024. As an application, for every odd $r\\ge1$ and every integer $k\\ge\\lceil r(3-2q_r)\\rceil$, we prove that the asymptotic minimum $(k-r)$-degree threshold for a tight Hamilton cycle in a $k$-uniform hypergraph is $q_r$, which settles, in this range, another conjecture of Lang and Sanhueza-Matamala in 2022 stating that the lower-bound construction due to Han and Zhao is asymptotically optimal. Our proof combines an isoperimetric inequality for Johnson graphs with a Kruskal-Katona-type theorem, which is of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38560",
  "author_total": 4,
  "authors": [
    {
      "name": "Hao Lin"
    },
    {
      "name": "Guanghui Wang"
    },
    {
      "name": "Jiande Zhang"
    },
    {
      "name": "Wenling Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5t78cteenpz42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38560",
  "fulltext_url": null,
  "id": "5t78cteenpz42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38560",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5t78cteenpz42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Large tight components and Dirac thresholds for tight Hamilton cycles",
  "title_text": "Large tight components and Dirac thresholds for tight Hamilton cycles",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5t78cteenpz42",
  "venue": "arXiv",
  "versions": []
}