{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R141",
  "content_hash": null,
  "slug": "cgr127-claim-exact-independence-number",
  "type": "claim",
  "title": "The independence number is 11",
  "summary": "An explicit independent 11-set meets an exhaustive colored branch-and-bound exclusion of every independent 12-set.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let\n\\[\nC=\\{z^3:z\\in\\mathbf F_{127}^{\\times}\\}\n\\]\nand join distinct \\(x,y\\) when \\(x-y\\in C\\). The set\n\\[\nW=\\{10,13,27,41,44,55,70,85,96,111,126\\}\n\\]\nis independent. Direct modular evaluation checks all \\(\\binom{11}{2}=55\\) pairs and finds no difference in \\(C\\), so \\(\\alpha(G)\\geq11\\).\n\nAn independent set in \\(G\\) is a clique in \\(\\overline G\\). The executable artifact `cgr127-artifact-exhaustive-size-12-search` performs a complete clique search in \\(\\overline G\\). Its greedy coloring bound is valid at every branch and the run finds no clique of order 12. Any independent set with more than 12 vertices would contain an independent 12-subset, so this proves \\(\\alpha(G)\\leq11\\). Therefore\n\\[\n\\boxed{\\alpha(G)=11}.\n\\]\n\nThis value was already known in computational Ramsey theory. Radziszowski's 2009 summary explicitly gives independence number 11 for the Hill-Irving graph \\(G_{127}\\). The independent artifact here supplies a short replay against the precise vertex and edge convention in the candidate.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets of the cubic-residue Cayley graph on Z/127Z",
    "bounds": {
      "modulus": {
        "min": 127,
        "max": 127
      },
      "vertices": {
        "min": 127,
        "max": 127
      },
      "set_size_checked": {
        "min": 11,
        "max": 12
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.cs.rit.edu/~spr/PUBL/dim09talk.pdf",
      "locator": "Stanislaw P. Radziszowski, Some Ramsey Problems Involving Triangles: Computational Approach, DIMACS Ramsey Theory Workshop slides, 28 May 2009, slide 33; independent certificate cgr127-artifact-exhaustive-size-12-search"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.cs.rit.edu/~spr/PUBL/dim09talk.pdf",
    "locator": "Stanislaw P. Radziszowski, Some Ramsey Problems Involving Triangles: Computational Approach, DIMACS Ramsey Theory Workshop slides, 28 May 2009, slide 33; independent certificate cgr127-artifact-exhaustive-size-12-search"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R143",
      "title": "The candidate defines a simple undirected 42-regular graph",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R142",
      "title": "This is a known Ramsey graph, not an open instance",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R140",
      "title": "Complete colored search excluding an independent 12-set",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "cubic-residue-graph-127-independence",
      "title": "cubic residue graph 127 independence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}