{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R788",
  "content_hash": null,
  "slug": "tlef15-attempt-literature-audit",
  "type": "attempt",
  "title": "The nearby literature counts and colors the triangles",
  "summary": "The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.",
  "relevance": "For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-attempt-literature-audit (“The nearby literature counts and colors the triangles”) documents a concrete method, search boundary, or failed route. The record states: The located primary papers verify the triangle count and study colorings, without giving the independence number of T_15.",
  "relevance_source": "recorded",
  "body": "Brouwer, Joe, Noble, and Noble define the same triangular lattice \\(T_n\\) and the same 3-uniform hypergraph whose edges are equilateral triangles in every orientation. Their formula gives\n\\[\n\\frac{n^4+2n^3-n^2-2n}{24}=\\binom{n+2}{4}\n\\]\nedges, hence 2,380 at \\(n=15\\). They determine or bound the minimum number of colors needed to avoid monochromatic equilateral triangles, including \\(f(15)\\leq5\\). Their paper does not report maximum single color-class sizes or \\(\\alpha(T_{15})\\).\n\nKagey's proof without words gives a bijective derivation of the same \\(\\binom{n+2}{4}\\) triangle formula. Searches keyed to the finite triangular lattice, equilateral-triangle-free sets, and hypergraph independence found no primary source settling the 15-row independence number. The certified interval in this entry should therefore be treated as a fresh bounded computation, with novelty still unverified.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "primary literature on equilateral triangles and forbidden monochromatic equilateral triangles in finite triangular lattices",
    "family": "finite triangular lattices T_n"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2405.12321",
      "locator": "Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321, abstract and Sections 1-2; Peter Kagey, A Proof Without Words: Triangles in the Triangular Grid, arXiv:2211.00186"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2405.12321",
    "locator": "Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321, abstract and Sections 1-2; Peter Kagey, A Proof Without Words: Triangles in the Triangular Grid, arXiv:2211.00186"
  },
  "models": [],
  "relations": [
    {
      "slug": "R789",
      "title": "The current certified interval is 33 through 56",
      "object_type": "claim",
      "relation": "contextualizes",
      "direction": "outgoing"
    },
    {
      "slug": "triangular-lattice-15-equilateral-free",
      "title": "triangular lattice 15 equilateral free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}