{
  "abstract": "For a $k$-uniform hypergraph $L$, let $h_L(n,q)$ be the minimum number of copies of $L$ in an $n$-vertex $k$-graph with $\\mathrm{ex}(n,L)+q$ edges. Let $t_L(n,q)$ be the minimum number of copies obtained by adding $q$ edges to an extremal $L$-free $k$-graph. A graph is color-critical if deleting a suitable edge decreases its chromatic number. Let $\\mathcal K_k(F)$ be the expanded graph obtained by adding $k-2$ new vertices to each edge of $F$, with disjoint sets of new vertices for distinct edges. Mubayi proved that, for every clique $K_r$ with $r\\ge4$ and fixed $ q>0$, $ h_{\\mathcal K_3(K_r)}(n,q)\\ge qt_{\\mathcal K_3(K_r)}(n,1)$ for every sufficiently large $n$. Pikhurko and Yilma later proved that, for every color-critical graph $F$, there is a constant $ε_F>0$ such that $h_{F}(n,q)=t_{F}(n,q)$ for every sufficiently large $n$ and $1\\le q\\leε_F n$. We extend this equality to expanded color-critical graphs. Let $d>k\\ge3$. For every color-critical graph $F$ with $χ(F)=d+1$, there is a constant $ε>0$ such that $ h_{\\mathcal K_k(F)}(n,q)=t_{\\mathcal K_k(F)}(n,q)$ for every sufficiently large $n$ and $1\\le q\\leεn^{k-1}$. Let $q_L(n)$ be the largest integer $q$ such that $h_L(n,s)=t_L(n,s)$ for every integer $1\\le s\\le q$. We give a sufficient condition for $q_L(n)=Θ_L(n^{k-1})$ and an explicit family satisfying it. We also give a different sufficient condition under which $q_L(n)=Ω_L(n^{k-1/2})$ and an explicit family satisfying it. \\noindent \\emph{Keywords.} Supersaturation; extremal graph; color-critical graph; chromatic number; expanded graph; hypergraph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15348",
  "author_total": 5,
  "authors": [
    {
      "name": "Liping Zhang"
    },
    {
      "name": "Qinghai Liu"
    },
    {
      "name": "Yanmei Hong"
    },
    {
      "name": "Wang Xi"
    },
    {
      "name": "Jinze Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25hsxehxpwkah.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15348",
  "fulltext_url": null,
  "id": "25hsxehxpwkah",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15348",
  "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": "W7221750817",
  "pdf_url": "https://arxiv.org/pdf/2610.15348",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25hsxehxpwkah.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.15348",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15348"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Supersaturation thresholds for expanded color-critical graphs",
  "title_text": "Supersaturation thresholds for expanded color-critical graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/25hsxehxpwkah",
  "venue": "arXiv",
  "versions": []
}