{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1706",
  "content_hash": null,
  "slug": "planted-clique-below-square-root-claim-literature-frontier",
  "type": "claim",
  "title": "Strongest checked neighboring result",
  "summary": "Polynomial-time recovery works at k on the order of √n, while low-degree, SoS, and several algorithm families have lower bounds below it.",
  "relevance": "Locates the present research frontier immediately below Polynomial-time recovery of planted cliques below the square-root scale.",
  "relevance_source": "recorded",
  "body": "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\nThis leaves the following boundary unresolved: No unrestricted polynomial-time algorithm or unconditional average-case lower bound is known for k=n^{1/2−δ}. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target.",
  "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": "R1707",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "planted-clique-below-square-root",
      "title": "planted clique below square root",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}