{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1707",
  "content_hash": null,
  "slug": "planted-clique-below-square-root-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Polynomial-time recovery works at k on the order of √n, while low-degree, SoS, and several algorithm families have lower bounds below it. Exact unresolved remainder: No unrestricted polynomial-time algorithm or unconditional average-case lower bound is known for k=n^{1/2−δ}.",
  "relevance": "This is the dated publication status for the canonical target Polynomial-time recovery of planted cliques below the square-root scale.",
  "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: Polynomial-time recovery works at k on the order of √n, while low-degree, SoS, and several algorithm families have lower bounds below it.\n\nThe exact unresolved remainder is: No unrestricted polynomial-time algorithm or unconditional average-case lower bound is known for k=n^{1/2−δ}.\n\nA complete resolution must meet the following acceptance conditions:\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": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1503.06447",
      "locator": "R. Meka, A. Potechin, and A. Wigderson, Sum-of-squares lower bounds for planted clique, STOC 2015. main SoS lower bounds"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1503.06447",
    "locator": "R. Meka, A. Potechin, and A. Wigderson, Sum-of-squares lower bounds for planted clique, STOC 2015. main SoS lower bounds"
  },
  "relations": [
    {
      "slug": "R1706",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1704",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1705",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "planted-clique-below-square-root",
      "title": "planted clique below square root",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}