{
  "schema_version": 1,
  "problem_number": "AMR-011-0031",
  "title": "A Cayley Graph with Chromatic Number Three and No Invariant Random Three-Coloring",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a connected seven-regular Cayley graph of (C5 * (C2 x C2)) x C2 with chromatic number three but no translation-invariant random proper three-coloring. A central involution partitions two finite subgroup graphs into pairs. Every proper three-coloring has at least four equal-colored pairs out of five in a blown-up five-cycle, and at most one out of four in a crown graph. Translation invariance would force the same probability to be at least 4/5 and at most 1/4. An explicit reduced-word rule gives a proper three-coloring of the entire graph, while a finite quotient supplies an invariant random four-coloring. Thus the invariant chromatic number is exactly four, giving a negative answer to the optimal invariant random coloring question of Lyons and Schramm, also recorded as Abért Question 31. Author: Alper Ferudun, Mercury Software GmbH. This English manuscript is an AI-assisted, self-audited and unrefereed preprint. The all-word proof is supplemented by an exact standard-library Python checker. No independent review, formal certification or absolute priority is claimed. Source record: AMR-011-0031 in ulamai/UnsolvedMath v1.6.0. Licensed under CC BY 4.0.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.GR",
    "math.PR"
  ],
  "keywords": [
    "Cayley graph",
    "invariant random coloring",
    "chromatic number",
    "free product",
    "central involution",
    "crown graph",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-011-0031/",
  "pdf_url": "https://eulersolve.org/papers/amr-011-0031/paper.pdf?v=475eeabaa63d",
  "doi": "10.5281/zenodo.23309994",
  "zenodo_record_url": "https://zenodo.org/records/23309994",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the optimal invariant random coloring question: the specified seven-regular Cayley graph has ordinary chromatic number three and invariant chromatic number four. The nonexistence argument excludes every translation-invariant Borel probability law, not only FIID or mixing constructions. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
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    "source.zip": {
      "sha256": "6742cda9f302effc062871def0719344995a2e235831a7d565a903da08e69419"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}