{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R546",
  "content_hash": null,
  "slug": "ngg-claim-thiele-construction",
  "type": "claim",
  "title": "Thiele gave the earlier linear construction",
  "summary": "Thiele's no-four-on-circle construction also enforces the no-three-in-line condition and yields more than \\(n/4\\) points.",
  "relevance": "For Grid points with no three collinear and no four concyclic, record ngg-claim-thiele-construction (“Thiele gave the earlier linear construction”) records a bound, answer, status fact, or structural consequence. The record states: Thiele's no-four-on-circle construction also enforces the no-three-in-line condition and yields more than \\(n/4\\) points.",
  "relevance_source": "recorded",
  "body": "Dong and Xu identify Thiele's 1995 paper as the earlier result for this exact combined planar condition. It gives \\(\\operatorname{ex}([n]^2;3,4)>n/4\\). Their newer construction raises the asymptotic coefficient to one third.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "Thiele's algebraically constructed planar grid subsets",
    "family": "finite planar grid subsets with no three collinear and no four concyclic"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0097-3165(95)90007-1",
      "locator": "Torsten Thiele, The no-four-on-circle problem, Journal of Combinatorial Theory Series A 71 (1995), pages 332-334"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0097-3165(95)90007-1",
    "locator": "Torsten Thiele, The no-four-on-circle problem, Journal of Combinatorial Theory Series A 71 (1995), pages 332-334"
  },
  "models": [],
  "relations": [
    {
      "slug": "R545",
      "title": "The published bounds leave a wide gap",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "no-three-collinear-no-four-concyclic-grid",
      "title": "no three collinear no four concyclic grid",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}