{
  "abstract": "We survey recent developments on minimum-degree thresholds in extremal graph theory, with an emphasis on structural refinements of the chromatic-threshold problem. The chromatic threshold asks when a linear minimum-degree condition forces bounded chromatic number in an \\(H\\)-free graph; its complete classification shows a striking finite trichotomy. We discuss several stronger and related threshold notions in which bounded colourability is replaced by bounded \\(H\\)-free homomorphic images, bounded neighbourhood complexity via VC-dimension, exact blowup structure for maximal \\(H\\)-free graphs, and improved quantitative bounds in graph removal lemmas. In contrast with the rigidity of the chromatic threshold, these newer parameters exhibit richer behaviour, including separations from \\(δ_χ(H)\\), new exact values, accumulation points in fixed-chromatic spectra, positivity phenomena for 3-chromatic graphs, asymmetric variants, and failures of monotonicity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39759",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinqi Huang"
    },
    {
      "name": "Hong Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn5djx5fax8w3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39759",
  "fulltext_url": null,
  "id": "bn5djx5fax8w3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39759",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39759",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn5djx5fax8w3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39759",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39759"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Minimum-degree thresholds beyond bounded colourability",
  "title_text": "Minimum-degree thresholds beyond bounded colourability",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/bn5djx5fax8w3",
  "venue": "arXiv",
  "versions": []
}