{
  "abstract": "For a fixed tree $T$ on $t$ vertices, let $\\ex(n,K_r,T)$ denote the maximum number of $r$-cliques in an $n$-vertex $T$-free graph. Write $n=α(t-1)+β$ with $0\\leβ<t-1$. A conjecture proposed in extremal-graph form by Alon and Shikhelman~\\cite{alon2016}, later stated explicitly as an exact clique-counting formula by Gerbner and Palmer~\\cite{GP2026}, asserts that \\[ \\ex(n,K_r,T)=α\\binom{t-1}{r}+\\binomβr, \\] with the extremal construction given by the vertex disjoint union of $α$ copies of $K_{t-1}$ and a graph with $β$ vertices. This paper proves this conjecture for every $n$ and for every tree $T$ whenever $C\\log t\\le r\\le t-1$ and $t$ is sufficiently large, where $C$ is an absolute constant. This lowers the previously known general threshold $r\\ge\\lceil(2t-1)/3\\rceil$, which is linear in $t$ to logarithmic order. The paper also proves the conjecture for every $3\\le r\\le t-1$ when at least $\\lceil(t-3)/2\\rceil$ leaves of $T$ have a common neighbor, and this extends the known results for stars and double stars.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25746",
  "author_total": 2,
  "authors": [
    {
      "name": "XiaoJun Zhao"
    },
    {
      "name": "YueJian Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edde2a56a675y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25746",
  "fulltext_url": null,
  "id": "edde2a56a675y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25746",
  "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": "W7204466536",
  "pdf_url": "https://arxiv.org/pdf/2608.25746",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edde2a56a675y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.25746",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25746"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Clique Counts in Graphs with a Forbidden Tree",
  "title_text": "Exact Clique Counts in Graphs with a Forbidden Tree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/edde2a56a675y",
  "venue": "arXiv",
  "versions": []
}