{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R142",
  "content_hash": null,
  "slug": "cgr127-claim-known-ramsey-graph",
  "type": "claim",
  "title": "This is a known Ramsey graph, not an open instance",
  "summary": "The graph dates to Hill and Irving's 1982 group partition, and later sources explicitly list its independence number as 11.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Hill and Irving studied the cubic partition of \\(\\mathbf Z_{127}^{\\times}\\) in 1982 and obtained the graph now usually denoted \\(G_{127}\\). It is a \\((4,12;127)\\)-Ramsey graph: it contains no \\(K_4\\) and its complement contains no \\(K_{12}\\). Radziszowski's 2009 computational Ramsey notes identify the same Cayley graph and explicitly list 127 vertices, 2,667 edges, degree 42, and independence number 11.\n\nSu, Li, Luo, and Li later developed an algorithm for cubic-residue graphs and their complements. Their 2002 paper reports computations for primes of the form \\(6m+1\\), and a 2017 account states that their clique-number computations extend through prime 1327. This places the parameter 127 well inside the previously computed range.\n\nGeneralized Paley clique bounds treat a single power-residue class. Here \\(\\alpha(G)=\\omega(\\overline G)\\), while \\(\\overline G\\) is the union of the other two cubic-character classes. The Ramsey-graph literature is therefore the direct exact source for this candidate.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the Hill-Irving cubic-residue graph G_127",
    "bounds": {
      "modulus": {
        "min": 127,
        "max": 127
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/S0195-6698(82)80007-3",
      "locator": "R. Hill and R. W. Irving, On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers, European Journal of Combinatorics 3 (1982), 35-50; Stanislaw P. Radziszowski, Some Ramsey Problems Involving Triangles, 2009, slide 33; W. Su, Q. Li, H. Luo, and G. Li, Discrete Mathematics 250 (2002), 197-209"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/S0195-6698(82)80007-3",
    "locator": "R. Hill and R. W. Irving, On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers, European Journal of Combinatorics 3 (1982), 35-50; Stanislaw P. Radziszowski, Some Ramsey Problems Involving Triangles, 2009, slide 33; W. Su, Q. Li, H. Luo, and G. Li, Discrete Mathematics 250 (2002), 197-209"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R141",
      "title": "The independence number is 11",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R140",
      "title": "Complete colored search excluding an independent 12-set",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "cubic-residue-graph-127-independence",
      "title": "cubic residue graph 127 independence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}