{
  "abstract": "The Erdős-Rogers function $f_{\\ell,s}(n)$ is the largest order of a $K_\\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\\ell+1$, the case where $s$ is much larger than $\\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $f_{\\ell,s}(n)\\geq n^{\\frac{\\ell}{2s}+O_\\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\\ell,s}(n)\\leq n^{\\frac{\\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\\ell,s}(n)= n^{\\frac{\\ell}{2s}+O_\\ell(s^{-2})}$. More precisely, we prove that for all $2\\leq \\ell<s$, we have $f_{\\ell,s}(n)\\leq n^{\\frac{\\ell}{2s-\\ell}+o(1)}$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\\ell=2$ case of our result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37987",
  "author_total": 3,
  "authors": [
    {
      "name": "Domagoj Bradač"
    },
    {
      "name": "Oliver Janzer"
    },
    {
      "name": "Rik Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ptxbcefqwmhz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37987",
  "fulltext_url": null,
  "id": "3ptxbcefqwmhz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37987",
  "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": "W7214989240",
  "pdf_url": "https://arxiv.org/pdf/2609.37987",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ptxbcefqwmhz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37987",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37987"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the asymptotics of the Erdős-Rogers function",
  "title_text": "On the asymptotics of the Erdős-Rogers function",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/3ptxbcefqwmhz",
  "venue": "arXiv",
  "versions": []
}