{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R545",
  "content_hash": null,
  "slug": "ngg-claim-known-asymptotic-bounds",
  "type": "claim",
  "title": "The published bounds leave a wide gap",
  "summary": "Current results give \\(n/3-o(n)\\leq f(n)\\leq2n\\).",
  "relevance": "For Grid points with no three collinear and no four concyclic, record ngg-claim-known-asymptotic-bounds (“The published bounds leave a wide gap”) records a bound, answer, status fact, or structural consequence. The record states: Current results give \\(n/3-o(n)\\leq f(n)\\leq2n\\).",
  "relevance_source": "recorded",
  "body": "Dong and Xu write this function as \\(\\operatorname{ex}([n]^2;3,4)\\): a set contains no three collinear points and no four concyclic points. Their Theorem 3, specialized to dimension two, gives \\(f(n)\\geq n/3-o(n)\\). The elementary upper bound \\(f(n)\\leq2n\\) follows by looking at the \\(n\\) horizontal rows. The proposed formula \\(2n-2\\) remains well above the best published general construction located in this audit.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the extremal function on every n by n integer grid"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2506.18113",
      "locator": "Zichao Dong and Zijian Xu, Large grid subsets without many cospherical points, pages 2-3, Theorem 3"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2506.18113",
    "locator": "Zichao Dong and Zijian Xu, Large grid subsets without many cospherical points, pages 2-3, Theorem 3"
  },
  "models": [],
  "relations": [
    {
      "slug": "R546",
      "title": "Thiele gave the earlier linear construction",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R542",
      "title": "Exact search through the 7 by 7 grid",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "no-three-collinear-no-four-concyclic-grid",
      "title": "no three collinear no four concyclic grid",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}