{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1432",
  "content_hash": null,
  "slug": "common-chromatic-subgraph-aleph-one-claim-literature-frontier",
  "type": "claim",
  "title": "Strongest checked neighboring result",
  "summary": "Every \\(\\aleph_1\\)-chromatic graph contains all sufficiently large odd cycles, so a common subgraph of chromatic number three is guaranteed.",
  "relevance": "Locates the present research frontier immediately below A common high-chromatic subgraph of two \\(\\aleph_1\\)-chromatic graphs.",
  "relevance_source": "recorded",
  "body": "Every \\(\\aleph_1\\)-chromatic graph contains all sufficiently large odd cycles, so a common subgraph of chromatic number three is guaranteed.\n\nThis leaves the following boundary unresolved: The first unknown guaranteed chromatic number is four; the countably infinite target is stronger. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/62",
      "locator": "Thomas F. Bloom, Erdős Problem #62, Erdős Problems database (living entry), accessed 2026-08-01. Problem #62, OPEN banner, statement, remarks, and bibliography"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/62",
    "locator": "Thomas F. Bloom, Erdős Problem #62, Erdős Problems database (living entry), accessed 2026-08-01. Problem #62, OPEN banner, statement, remarks, and bibliography"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1433",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "common-chromatic-subgraph-aleph-one",
      "title": "common chromatic subgraph aleph one",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}