{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1705",
  "content_hash": null,
  "slug": "planted-clique-below-square-root-attempt-open-remainder",
  "type": "attempt",
  "title": "Work at the unresolved boundary",
  "summary": "No unrestricted polynomial-time algorithm or unconditional average-case lower bound is known for k=n^{1/2−δ}.",
  "relevance": "Turns the remaining uncertainty in Polynomial-time recovery of planted cliques below the square-root scale into a checkable research target.",
  "relevance_source": "recorded",
  "body": "Research should address this boundary directly: No unrestricted polynomial-time algorithm or unconditional average-case lower bound is known for k=n^{1/2−δ}.\n\nA claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations:\n- For some fixed δ>0, give and prove a polynomial-time exact-recovery algorithm.\n- Or prove polynomial-time recovery impossible under an explicit unconditional average-case model, which may require a new lower-bound framework.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/1503.06447",
      "locator": "TheoremDB editorial route recorded 2026-08-01"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1503.06447",
    "locator": "TheoremDB editorial route recorded 2026-08-01"
  },
  "relations": [
    {
      "slug": "R1707",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "planted-clique-below-square-root",
      "title": "planted clique below square root",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}