{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1420",
  "content_hash": null,
  "slug": "c4-supersaturation-at-extremal-threshold-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved when \\(n=q^2+q+1\\) for an even integer \\(q\\); the general orders remain open. Exact unresolved remainder: Establish the \\(\\Omega(\\sqrt n)\\) count uniformly for all sufficiently large \\(n\\), or refute it.",
  "relevance": "This is the dated publication status for the canonical target Four-cycle supersaturation just above the extremal threshold.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: The conjecture is proved when \\(n=q^2+q+1\\) for an even integer \\(q\\); the general orders remain open.\n\nThe exact unresolved remainder is: Establish the \\(\\Omega(\\sqrt n)\\) count uniformly for all sufficiently large \\(n\\), or refute it.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove a universal \\(c>0\\) and threshold \\(n_0\\) giving the bound for every \\(n\\ge n_0\\), or construct an infinite counterexample family with \\(o(\\sqrt n)\\) four-cycles.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/60",
      "locator": "Thomas F. Bloom, Erdős Problem #60, Erdős Problems database (living entry), accessed 2026-08-01. Problem #60, OPEN banner, statement, remarks, and bibliography"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/60",
    "locator": "Thomas F. Bloom, Erdős Problem #60, Erdős Problems database (living entry), accessed 2026-08-01. Problem #60, OPEN banner, statement, remarks, and bibliography"
  },
  "relations": [
    {
      "slug": "R1419",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1417",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1418",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "c4-supersaturation-at-extremal-threshold",
      "title": "c4 supersaturation at extremal threshold",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}