{
  "abstract": "We prove that there is an absolute constant $c>0$ such that every graph of chromatic number at least $r$ and at most $cr^3\\log^2 r$ edges contains at least $\\binom r3$ triangles. The proof has three ingredients. First, a sparse-core argument based on a triangle-sensitive coloring estimate of Harris extracts, from any counterexample, an induced subgraph of order $O(r)$ and chromatic number at least $(1-η)r$. Second, we prove an order-uniform stability theorem: for every $β>0$ there is $γ>0$, independent of the constant in the linear order bound, such that every sufficiently large $s$-critical graph $J$ of order $O(s)$ with $ω(J)\\le (1-β)s$ has at least $\\binom s3+γs^3$ triangles. The proof combines the excess method and the modified-independent-set argument of Fox, Tidor, and Zhang. Third, we prove the exact bound when the graph contains a clique of order at least $(1-δ)r$. This uses a new dense common-palette list analogue of Harris's edge--triangle estimate: if every list occupies a fixed positive proportion of a common palette, then the edge-triangle coloring bound survives up to a constant factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26247",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuyan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/79mzznmp33st4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26247",
  "fulltext_url": null,
  "id": "79mzznmp33st4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26247",
  "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": "W7214113241",
  "pdf_url": "https://arxiv.org/pdf/2609.26247",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/79mzznmp33st4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26247",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26247"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sparse chromatic graphs and the complete-graph triangle bound",
  "title_text": "Sparse chromatic graphs and the complete-graph triangle bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/79mzznmp33st4",
  "venue": "arXiv",
  "versions": []
}