{
  "abstract": "A graph $H$ is $K_{r+1}$-Turán-good if, for every sufficiently large $n$, the Turán graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs, and it is strictly $K_{r+1}$-Turán-good when this extremal graph is unique. Morrison, Nir, Norin, Rzążewski and Wesolek proved that for every graph $H$ with at least one edge, when $r\\ge 300v(H)^9$, $H$ is $K_{r+1}$-Turán-good. They asked whether the above bound could be reduced to quadratic order in $v(H)$. In this paper, we answer their question affirmatively, and give a stronger result. We prove that there is an absolute constant $C>0$ such that every graph $H$ with at least one edge is strictly $K_{r+1}$-Turán-good whenever $r\\ge Cv(H)$. Our proof uses a substantially different cluster method based on polymer models. Besides proving the vertex-linear Turán-good threshold, this method appears to have further applications. As one illustration, we prove that for every graph $H$ with at least one edge, the normalized chromatic polynomial $P_H(x)/x^{v(H)}$ is strictly increasing for every real $x\\ge CΔ(H)$, which strengthens the previously best result where $x\\ge CΔ(H)^{3/2}$. This proves a conjecture of Fadnavis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25038",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiamiao Zhao"
    },
    {
      "name": "Liying Kang"
    },
    {
      "name": "Yuanpei Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/95sznas9s0hqg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25038",
  "fulltext_url": null,
  "id": "95sznas9s0hqg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25038",
  "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": "W7214119980",
  "pdf_url": "https://arxiv.org/pdf/2609.25038",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/95sznas9s0hqg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25038",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25038"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Vertex-Linear Threshold for Eventually Turán good and the Cluster Method",
  "title_text": "A Vertex-Linear Threshold for Eventually Turán good and the Cluster Method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/95sznas9s0hqg",
  "venue": "arXiv",
  "versions": []
}