{
  "abstract": "We prove that the generalized threshold Ramsey number R(5,10;5,8) equals 21. This number corresponds to the set Ramsey number r_5(10,8) introduced by Harborth. The proof is computer-assisted. We first enumerate all (4,6;5,8)-good and (5,10;4,4)-good graphs. Then we apply the lucky-vertex inequality derived from the McKay--Radziszowski edge identity to filter candidate pairs for a gluing search, and encode the remaining cases as SAT instances. The search yields exactly 17 non-isomorphic (5,10;5,8;20)-good graphs. Finally, we exclude all one-vertex extensions of these 17 graphs, which proves that no (5,10;5,8;21)-good graph exists.Additionally, we update the tabulation of small set Ramsey numbers r_5(m,n) for 1 <= m,n <= 10.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29584",
  "author_total": 2,
  "authors": [
    {
      "name": "Wancai Guo"
    },
    {
      "name": "Min Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94116mnnxb6zs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29584",
  "fulltext_url": null,
  "id": "94116mnnxb6zs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29584",
  "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": "W7223240853",
  "pdf_url": "https://arxiv.org/pdf/2610.29584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94116mnnxb6zs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Computer-Assisted Proof that R(5,10;5,8)=21 for the Generalized Threshold Ramsey Number",
  "title_text": "A Computer-Assisted Proof that R(5,10;5,8)=21 for the Generalized Threshold Ramsey Number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/94116mnnxb6zs",
  "venue": "arXiv",
  "versions": []
}