{
  "abstract": "In 1991, Erdős, Faudree, Rousseau and Schelp asked to estimate the smallest $u=u(n,d,s)$ such that every $n$-vertex graph with average degree at least $d$ has a subgraph on at most $u$ vertices with minimum degree at least $s$. We resolve this problem up to polylogarithmic factors for all even values of $s$. More precisely, we prove that every $n$-vertex graph with minimum degree $d\\geq s$ has a subgraph on $O_s\\left(\\max\\left\\{\\frac{n}{d^{s/(s-2)}}(\\log n)^{180},\\log n\\right\\}\\right)$ vertices with minimum degree at least $s$, whenever $s\\ge 4$ is even. This improves a result of Janzer, Sudakov and Tomon, who obtained a similar conclusion for subgraphs of average degree at least $s$. We also prove that for every positive integer $s$ and every $\\varepsilon>0$, there exists a finite family $\\mathcal{H}$ of graphs with minimum degree $s$ such that $\\operatorname{ex}(n,\\mathcal{H})=O(n^{2-2/s+\\varepsilon})$. This is tight and is the first result of similar kind that applies to odd values of $s$ greater than 3.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26763",
  "author_total": 4,
  "authors": [
    {
      "name": "Zach Hunter"
    },
    {
      "name": "Barnabás Janzer"
    },
    {
      "name": "Oliver Janzer"
    },
    {
      "name": "Benny Sudakov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r5zpe0m88s7a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26763",
  "fulltext_url": null,
  "id": "8r5zpe0m88s7a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26763",
  "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": "W7223015186",
  "pdf_url": "https://arxiv.org/pdf/2610.26763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r5zpe0m88s7a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.26763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Small subgraphs of large minimum degree",
  "title_text": "Small subgraphs of large minimum degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/8r5zpe0m88s7a",
  "venue": "arXiv",
  "versions": []
}