{ "schemaVersion": "1.1", "slug": "z20-equals-6", "title": "z(20) = 6: resolving the first open case of Erdős problem 758", "shortTitle": "z(20) = 6 (Erdős problem 758)", "url": "https://evidence-press.pages.dev/releases/z20-equals-6/", "oneLine": "Every graph on 20 vertices can be split into six parts, each a clique or an independent set — and 6 is best possible. A computer-assisted proof, released with its full certificate chain, settles the first open case of an Erdős–Gimbel problem.", "abstract": "How few colours do you need to partition any 20-vertex graph so that every colour class is either a clique (all pairs connected) or an independent set (no pairs connected)? The answer for up to 19 vertices was known; at 20 it was either 6 or 7. This computer-assisted proof establishes that the answer is 6. The lower bound is a short, hand-checkable argument about the Paley graph on 17 vertices; the upper bound reduces all possible counterexamples to two SAT problems whose impossibility is certified with proof objects checked by four independent checkers, including a formally verified one.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "candidate-2026-07-26", "doi": "10.5281/zenodo.21647645", "doiUrl": "https://doi.org/10.5281/zenodo.21647645", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/z20-cochromatic/releases/download/candidate-2026-07-26/z20_equals_6_proof.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/z20-cochromatic/master/paper/z20_equals_6_proof.pdf", "zenodoUrl": "https://zenodo.org/records/21647645", "repoUrl": "https://github.com/ipitchford/z20-cochromatic", "releaseUrl": "https://github.com/ipitchford/z20-cochromatic/releases/tag/candidate-2026-07-26", "markdownUrl": "https://evidence-press.pages.dev/releases/z20-equals-6/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/z20-equals-6/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/z20-equals-6.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/z20-equals-6.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/z20-equals-6.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 proof. No human mathematician has verified the complete argument, and there is no independent external reproduction or end-to-end proof-assistant formalisation. A dated replay on one machine (macOS/arm64) covered catalogue regeneration, cocolourability checks, CNF regeneration, and multiple proof-checker validations. Encoding soundness rests on semantic spot-testing, not proof, so implementation errors remain logically possible. The mathematics was generated by AI systems with human problem selection and 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": "Erdős problem 758 (Erdős–Gimbel, cochromatic numbers of small graphs)", "url": "https://www.erdosproblems.com/758" }, "keywords": [ "cochromatic number", "Erdős problem 758", "Erdős–Gimbel", "Ramsey theory", "R(4,4)", "Paley graph", "SAT certificates", "DRUP", "LRAT", "cake_lpr", "computer-assisted proof", "AI-generated mathematics" ], "keyResults": [ "z(20) = 6: the maximum cochromatic number over all 20-vertex graphs is exactly 6 (previously known to be 6 or 7).", "Lower bound, hand-checkable: the Paley graph on 17 vertices has no clique or independent set of size 4, so its cochromatic number is at least ⌈17/3⌉ = 6.", "Upper bound: any 20-vertex counterexample reduces, via R(4,4) = 18, to cases built around the two 16-vertex (4,4)-Ramsey graphs; the resulting two CNF formulas (64 variables, 104,524 clauses each) are unsatisfiable, with DRUP and LRAT certificates.", "Supporting recomputations: R(4,4) = 18, regeneration of the (4,4)-Ramsey graph catalogue, and the known values z(8) ≤ 3 and z(12) ≤ 4." ], "evidencePackage": "Two SAT instances with DRUP and LRAT unsatisfiability certificates, checked by a custom Python RUP checker, a C RUP checker, drat-trim, and the formally verified cake_lpr; a hand-verifiable Paley(17) lower-bound argument; a 13-page proof PDF; a machine-readable AI_INDEX; SHA-256 manifests; portable replay script.", "openProblems": [ "Independently reproduce the reduction: regenerate the (4,4)-Ramsey catalogue and the two core CNFs from a fresh implementation, and confirm unsatisfiability with independent tooling.", "Formally verify the encoding: the LRAT certificates are already checked by a formally verified checker (cake_lpr), but the reduction from 'z(20) ≤ 6' to the two CNFs is not itself machine-checked — formalising it in Lean or Isabelle would close the main gap.", "Extend the method to z(n) for n = 21 and beyond, where the Erdős–Gimbel table is open.", "Study whether the Paley(17)-based lower-bound construction generalises to give improved lower bounds for larger n.", "Use the release as a benchmark case for AI-generated mathematics workflows: claim–evidence indexing, certificate exchange, and independent agent reproduction." ], "relatedWorks": [ { "citation": "Erdős problem 758 (Erdős & Gimbel): determine z(n) for small n — records the known values through n = 19 and Mehta's computational confirmation that z(12) = 4.", "url": "https://www.erdosproblems.com/758" }, { "citation": "Companion release: VR2(K4) = 20, which reuses this paper's upper-bound proof objects.", "url": "https://doi.org/10.5281/zenodo.21647654" }, { "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 — lists the two 16-vertex (4,4)-Ramsey graphs used in the reduction.", "url": "https://users.cecs.anu.edu.au/~bdm/data/ramsey.html" }, { "citation": "Tan, Y. K., Heule, M. J. H., & Myreen, M. O. cake_lpr: a verified LRAT proof checker (CakeML) — the formally verified checker used on the certificates.", "url": "https://github.com/tanyongkiam/cake_lpr" } ] }