{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R520",
  "content_hash": null,
  "slug": "mgsl-claim-moore-upper-bound",
  "type": "claim",
  "title": "The Moore bound gives a universal ceiling of 24",
  "summary": "Girth 25 would require 1,062,881 vertices, which is 32,681 too many.",
  "relevance": "For Largest girth from two generators of SL(2,101), record mgsl-claim-moore-upper-bound (“The Moore bound gives a universal ceiling of 24”) records a bound, answer, status fact, or structural consequence. The record states: Girth 25 would require 1,062,881 vertices, which is 32,681 too many.",
  "relevance_source": "recorded",
  "body": "Suppose a 4-regular graph has girth at least 25. The nonbacktracking paths of length at most 12 starting at a fixed vertex have distinct endpoints. Otherwise two such paths would give a cycle of length at most 24. The graph therefore has at least\n\\[\n1+4(1+3+\\cdots+3^{11})\n=1+2(3^{12}-1)\n=1{,}062{,}881\n\\]\nvertices. The Cayley graphs in this problem have 1,030,200 vertices, so their girth is at most 24.",
  "status": "established",
  "evidence_grade": "proved",
  "scope": {
    "kind": "bounded",
    "statement": "every simple 4-regular graph on 1,030,200 vertices, including every eligible Cayley graph in the problem",
    "bounds": {
      "vertices": {
        "min": 1030200,
        "max": 1030200
      },
      "degree": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0707.1833",
      "locator": "Gamburd, Hoory, Shahshahani, Shalev, and Virag, On the girth of random Cayley graphs, Section 1 discusses the Moore counting bound; the numerical specialization is shown here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0707.1833",
    "locator": "Gamburd, Hoory, Shahshahani, Shalev, and Virag, On the girth of random Cayley graphs, Section 1 discusses the Moore counting bound; the numerical specialization is shown here"
  },
  "models": [],
  "relations": [
    {
      "slug": "R519",
      "title": "The maximum girth is currently certified between 17 and 24",
      "object_type": "claim",
      "relation": "bounds",
      "direction": "outgoing"
    },
    {
      "slug": "max-girth-sl2-101-generators",
      "title": "max girth sl2 101 generators",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}