{
  "abstract": "Let $T_r(n)$ denote the complete $r$-partite graph on $n$ vertices whose part sizes differ by at most one. A graph $H$ is called $K_{r+1}$-Turán-good if, for all sufficiently large $n$, the graph $T_r(n)$ contains the maximum number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. We say that $H$ is $K_{r+1}$-Turán-stable if, for every $\\varepsilon>0$, there exist $δ>0$ and $n_0$ such that, whenever $n\\ge n_0$ and an $n$-vertex $K_{r+1}$-free graph $G$ contains at least $\\mathrm{ex}(n,H,K_{r+1})-δn^{v(H)}$ copies of $H$, the edit distance between $G$ and $T_r(n)$ is at most $\\varepsilon n^2$, where $v(H)$ denotes the order of $H$. In this paper, we prove that every graph $H$ is both $K_{r+1}$-Turán-good and $K_{r+1}$-Turán-stable for $r\\ge 4v(H)^3+11v(H)^2$. This strengthens results of Morrison, Nir, Norin, Rzążewski, and Wesolek~[J. Combin. Theory Ser. B, 2023] and Gerbner and Hama Karim~[J. Graph Theory, 2024], and gives a positive answer to a question of Morrison, Nir, Norin, Rzążewski, and Wesolek. We also prove that $\\inj(H,G)\\le \\inj(H,T_r(n))$ for every $n$-vertex $K_{r+1}$-free graph $G$ and for $r\\ge 40v(H)^3$, where $\\inj(H,G)$ denotes the number of injective homomorphisms from $H$ to $G$. Finally, we give a negative answer to another question of Morrison, Nir, Norin, Rzążewski, and Wesolek.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29493",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiabao Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b49wv10jby0dg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29493",
  "fulltext_url": null,
  "id": "b49wv10jby0dg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29493",
  "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": "W7214302171",
  "pdf_url": "https://arxiv.org/pdf/2609.29493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b49wv10jby0dg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Every graph is eventually Turán-good and Turán-stable",
  "title_text": "Every graph is eventually Turán-good and Turán-stable",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/b49wv10jby0dg",
  "venue": "arXiv",
  "versions": []
}