{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R534",
  "content_hash": null,
  "slug": "mols10-claim-currently-open",
  "type": "claim",
  "title": "Existence of three MOLS of order 10 remains open",
  "summary": "A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem.",
  "relevance": "For Three mutually orthogonal Latin squares of order ten, record mols10-claim-currently-open (“Existence of three MOLS of order 10 remains open”) records a bound, answer, status fact, or structural consequence. The record states: A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem.",
  "relevance_source": "recorded",
  "body": "Let \\(N(10)\\) be the largest number of pairwise orthogonal Latin squares of order 10. Parker's 1959 construction gives \\(N(10)\\geq2\\). Bright, Keita, and Stevens state in their 2026 paper that it is unknown whether \\(N(10)\\geq3\\), and call the construction or exclusion of 3 MOLS(10) a longstanding open problem.\n\nTheir exhaustive SAT results address a restricted hypothesis: one square in the proposed triple contains a \\(4\\times4\\) Latin subsquare. Myrvold classified 28 possible mate-pattern pairs under that hypothesis. The SAT investigation verifies the absence of an orthogonal pair in 20 pattern cases and constructs an orthogonal pair in each of the other eight. Those eight cases still require analysis of the third square. The general three-square question remains undecided.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the reported existence status of three mutually orthogonal Latin squares of order 10, checked on 2026-07-24",
    "bounds": {
      "order": {
        "min": 10,
        "max": 10
      },
      "square_count": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.37236/13960",
      "locator": "Curtis Bright, Amadou Keita, and Brett Stevens, Myrvold's Results on Orthogonal Triples of 10 x 10 Latin Squares: A SAT Investigation, Electronic Journal of Combinatorics 33(1) (2026), P1.30, DOI 10.37236/13960; abstract and pages 1-3 of arXiv:2503.10504v2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.37236/13960",
    "locator": "Curtis Bright, Amadou Keita, and Brett Stevens, Myrvold's Results on Orthogonal Triples of 10 x 10 Latin Squares: A SAT Investigation, Electronic Journal of Combinatorics 33(1) (2026), P1.30, DOI 10.37236/13960; abstract and pages 1-3 of arXiv:2503.10504v2"
  },
  "relations": [
    {
      "slug": "R535",
      "title": "The published near triple contains exactly two orthogonal pairs",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R533",
      "title": "Exact replay of the order-10 near triple",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "three-mols-order-10",
      "title": "three mols order 10",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}