{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1166",
  "content_hash": null,
  "slug": "triangle-free-degree-six-five-colouring-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved. Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.",
  "relevance": "For triangle free degree six five colouring, pins the dated research frontier: UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved. Give a proof producing a.",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. The MathOverflow page remains open with zero answers. A 2023 paper proves the claim for maximal triangle-free graphs of maximum degree below seven and verifies bounded orders, while describing the unrestricted Reed-conjecture case as unresolved.\n\nA complete resolution must satisfy this condition: Give a proof producing a proper five-colouring for every finite simple triangle-free graph of maximum degree at most six, or exhibit a finite triangle-free graph with maximum degree at most six and chromatic number at least six.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/37923/does-every-triangle-free-graph-with-maximum-degree-at-most-6-have-a-5-colouring",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/37923/does-every-triangle-free-graph-with-maximum-degree-at-most-6-have-a-5-colouring",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "relations": [
    {
      "slug": "R1165",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "triangle-free-degree-six-five-colouring",
      "title": "triangle free degree six five colouring",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}