{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R644",
  "content_hash": null,
  "slug": "r55-attempt-order-43-witness-audit",
  "type": "attempt",
  "title": "The order-43 witness remains missing",
  "summary": "The candidate cites a 42-vertex collection, while current sources retain 43 as the best lower bound.",
  "relevance": "For A 43-vertex graph for the diagonal Ramsey problem R(5,5), record r55-attempt-order-43-witness-audit (“The order-43 witness remains missing”) documents a concrete method, search boundary, or failed route. The record states: The candidate cites a 42-vertex collection, while current sources retain 43 as the best lower bound.",
  "relevance_source": "recorded",
  "body": "The candidate contains no graph6 record or adjacency matrix of order 43. Its computation identifies the first record in `r55_42some.g6`. The leading graph6 byte confirms order 42.\n\nMcKay and Radziszowski reported 656 known \\((5,5,42)\\)-graphs in section 4 of their 1997 paper. They checked that those graphs could not be extended by one vertex and conjectured both \\(R(5,5)=43\\) and completeness of the 656-graph collection. The completeness statement remains a conjecture in that paper, so the failed extensions do not exclude every possible 43-vertex graph.\n\nAngeltveit and McKay's 2026 paper states that Exoo's lower bound 43 remains the best and proves the current upper bound 46. McKay's data page likewise says that graphs with 43 through 47 vertices could exist. The current interval is\n\\[\n43\\leq R(5,5)\\leq46.\n\\]\nAn explicit \\((5,5,43)\\)-graph would improve its lower endpoint to 44. No such artifact was supplied or located in the checked primary sources.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the candidate record and current primary sources checked for an explicit Ramsey(5,5,43) graph through 2026-07-24",
    "bounds": {
      "requested_vertex_count": {
        "min": 43,
        "max": 43
      },
      "search_year": {
        "min": 2026,
        "max": 2026
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1002/jgt.70029",
      "locator": "Vigleik Angeltveit and Brendan D. McKay, R(5,5) at most 46, Journal of Graph Theory 112 (2026), Introduction and Theorem 1.1; Brendan McKay and Stanislaw Radziszowski, Subgraph Counting Identities and Ramsey Numbers, Journal of Combinatorial Theory B 69 (1997), section 4, pages 9-10; McKay's Combinatorial Data page, section The largest known Ramsey(5,5)-graphs; checked 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1002/jgt.70029",
    "locator": "Vigleik Angeltveit and Brendan D. McKay, R(5,5) at most 46, Journal of Graph Theory 112 (2026), Introduction and Theorem 1.1; Brendan McKay and Stanislaw Radziszowski, Subgraph Counting Identities and Ramsey Numbers, Journal of Combinatorial Theory B 69 (1997), section 4, pages 9-10; McKay's Combinatorial Data page, section The largest known Ramsey(5,5)-graphs; checked 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R645",
      "title": "The cited public graph certifies R(5,5) at least 43",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "ramsey-55-43-graph",
      "title": "ramsey 55 43 graph",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}