{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R123",
  "content_hash": null,
  "slug": "cdg-claim-ladder-terminology",
  "type": "claim",
  "title": "Polynomial value sets are chromatically intersective",
  "summary": "Positive values of an integer-valued polynomial with zero constant term form a ladder, the distance-set condition equivalent to infinite chromatic number.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Guerreiro, Ruzsa, and Silva call a set \\(D\\) a ladder when every finite coloring of the integers contains arbitrarily long monochromatic \\(D\\)-paths. Their Corollary 4 includes the positive values of every integer-valued polynomial \\(p\\) satisfying \\(p(0)=0\\). Section 5 relates the ladder and chromatic-intersectivity language to the integer distance graph \\(G_D\\). Taking \\(p(n)=n^3\\) places the database entry inside this general result.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "positive values of every integer-valued polynomial p with p(0)=0"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1606.00418",
      "locator": "Guerreiro, Ruzsa, and Silva, Monochromatic Paths for the Integers, Corollary 4 and section 5"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1606.00418",
    "locator": "Guerreiro, Ruzsa, and Silva, Monochromatic Paths for the Integers, Corollary 4 and section 5"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R122",
      "title": "The cube-distance graph has countably infinite chromatic number",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "chromatic-number-cube-distance-graph",
      "title": "chromatic number cube distance graph",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}