{
  "abstract": "Let $δ^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\\ge2$, $r\\ge k+1$ and $\\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(δ^*_{k,r}+\\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p \\ge C(\\log n/n^{r-k})^{1/(\\binom{r}{k}-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00924",
  "author_total": 1,
  "authors": [
    {
      "name": "Tuan Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12dfekgp5nav5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00924",
  "fulltext_url": null,
  "id": "12dfekgp5nav5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00924",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215987467",
  "pdf_url": "https://arxiv.org/pdf/2610.00924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12dfekgp5nav5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The threshold for fractional clique decompositions of random hypergraphs, via matrix scaling",
  "title_text": "The threshold for fractional clique decompositions of random hypergraphs, via matrix scaling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/12dfekgp5nav5",
  "venue": "arXiv",
  "versions": []
}