{
  "schemaVersion": "1.1",
  "slug": "vr2-k4-equals-20",
  "title": "VR2(K4) = 20: twenty vertices force two vertex-disjoint monochromatic K4s",
  "shortTitle": "VR2(K4) = 20",
  "url": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/",
  "oneLine": "Colour every edge among 20 points red or blue and you cannot avoid two completely separate single-colour foursomes — while a carefully built 19-point colouring can. A companion result to z(20) = 6.",
  "abstract": "The classical Ramsey number R(4,4) = 18 says that among 18 points with red/blue connections there is always a single-colour foursome. This release asks for more: how many points guarantee two such foursomes sharing no points at all? The claimed answer is exactly 20. A hand-checkable 19-vertex colouring (a 'Paley twin') has no two disjoint monochromatic K4s, while the impossibility of avoiding them at 20 vertices reuses the SAT certificates from the companion z(20) = 6 release.",
  "datePublished": "2026-07-28",
  "dateModified": "2026-07-28",
  "version": "0.1-candidate",
  "doi": "10.5281/zenodo.21647654",
  "doiUrl": "https://doi.org/10.5281/zenodo.21647654",
  "conceptDoi": null,
  "pdfUrl": "https://github.com/ipitchford/z20-cochromatic/releases/download/vr2-k4-v0.1-candidate/vr2_k4_equals_20_candidate.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/z20-cochromatic/master/applications/vr2-k4/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21647654",
  "repoUrl": "https://github.com/ipitchford/z20-cochromatic",
  "releaseUrl": "https://github.com/ipitchford/z20-cochromatic/releases/tag/vr2-k4-v0.1-candidate",
  "markdownUrl": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/vr2-k4-equals-20.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/vr2-k4-equals-20.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/vr2-k4-equals-20.svg",
  "media": [],
  "authors": [
    "GPT 5.6 Sol",
    "Claude Opus 4.8 / 5"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate result, marked as a prerelease. No independent external reproduction, peer review, or full formal verification is claimed. The lower-bound witness is directly inspectable and replayable; the upper bound inherits the assurance boundary of the z(20) = 6 release, including the fact that encoding soundness is spot-tested rather than proved. The definition and notation VR2(K4) are as given by the project; the release is authored by AI systems with human project mediation."
  },
  "provenance": {
    "aiGenerated": true,
    "generatedBy": [
      "GPT 5.6 Sol",
      "Claude Opus 4.8 / 5"
    ],
    "humanRole": "problem selection, mediation, and publication management",
    "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution."
  },
  "problem": {
    "name": "Vertex-disjoint Ramsey-type quantities (related context: Erdős problem 758)",
    "url": "https://www.erdosproblems.com/758"
  },
  "keywords": [
    "Ramsey theory",
    "vertex-disjoint monochromatic subgraphs",
    "VR2(K4)",
    "K4",
    "R(4,4)",
    "Paley graph",
    "SAT certificates",
    "LRAT",
    "cake_lpr",
    "computer-assisted proof",
    "AI-generated mathematics"
  ],
  "keyResults": [
    "VR2(K4) = 20: every red–blue colouring of the edges of K20 contains two vertex-disjoint monochromatic copies of K4.",
    "Sharpness: an explicit 19-vertex 'Paley-twin' colouring admits no two vertex-disjoint monochromatic K4s, so VR2(K4) > 19.",
    "The upper bound reuses the z(20) = 6 reduction and SAT unsatisfiability certificates (DRUP/LRAT, checked including by the formally verified cake_lpr)."
  ],
  "evidencePackage": "A human-readable candidate paper with TeX source; a hand-checkable 19-vertex lower-bound construction; a dependency-free direct verifier with its output included; reused DRUP/LRAT proof objects from the z20 release (pinned to a specific commit); an AI-readable claim/evidence index; SHA-256 manifest.",
  "openProblems": [
    "Verify the 19-vertex Paley-twin construction independently — it is small enough for a fresh implementation in an afternoon.",
    "Determine VR2(K5), or more generally VRk(Km) for small k and m, for which the same reduction-plus-SAT architecture is a plausible route.",
    "Establish whether VR2(K4) = 20 admits a certificate-free human proof, given that R(4,4) = 18 does.",
    "Search the literature for prior appearances of this quantity under other notation, and connect it to known results on vertex-disjoint monochromatic structures.",
    "Formalise the reduction linking the z(20) proof objects to the VR2 claim."
  ],
  "relatedWorks": [
    {
      "citation": "Companion release: z(20) = 6 (Erdős problem 758), whose upper-bound proof objects this result reuses.",
      "url": "https://doi.org/10.5281/zenodo.21647645"
    },
    {
      "citation": "Greenwood, R. E., & Gleason, A. M. (1955). Combinatorial relations and chromatic graphs. Canadian Journal of Mathematics, 7, 1–7 — establishes R(4,4) = 18.",
      "url": "https://doi.org/10.4153/CJM-1955-001-4"
    },
    {
      "citation": "McKay, B. Ramsey graphs data page — the two 16-vertex (4,4)-Ramsey graphs.",
      "url": "https://users.cecs.anu.edu.au/~bdm/data/ramsey.html"
    },
    {
      "citation": "Erdős problem 758 (related context).",
      "url": "https://www.erdosproblems.com/758"
    }
  ]
}