{
  "abstract": "The Erdős--Stone Theorem asserts that if a graph has edge density $1-1/r+δ$ then it contains a complete $(r+1)$-partite graph with $b$ vertices in each part, where $b=b_n(r,δ) \\gg 1$. The celebrated Chvátal--Szemerédi theorem determined the exact order of $b_n(r,δ)$ for every $δ< 1/r^3$. Their bound, however, is not tight when $δ=1/r-ε$, that is, when the graph has edge density $1-ε$ for small $ε$. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erdős--Stone problem for all edge densities. More precisely, we prove that for every integer $r\\geq 2$ and $0< δ< 1/r$ we have $$ b_n(r,δ)=Θ\\left(\\frac{\\log n}{(1/r-δ)r\\log(1/δ)}\\right)\\;. $$ The lower bound is obtained using a Kövari-Sós-Turán-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01498",
  "author_total": 2,
  "authors": [
    {
      "name": "Asaf Shapira"
    },
    {
      "name": "Raphael Yuster"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drwwgc496wa95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01498",
  "fulltext_url": null,
  "id": "drwwgc496wa95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01498",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7206153261",
  "pdf_url": "https://arxiv.org/pdf/2609.01498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drwwgc496wa95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Tight Erdős-Stone Bound for All Graph Densities",
  "title_text": "A Tight Erdős-Stone Bound for All Graph Densities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/drwwgc496wa95",
  "venue": "arXiv",
  "versions": []
}