{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1356",
  "content_hash": null,
  "slug": "projective-sets-ramsey-consistency-strength-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow answers give upper and lower consistency bounds and say the exact strength is open. Later checked problem lists continue to ask for the missing comparison, but the audit did not locate a definitive current survey.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 both MathOverflow answers and all comments were checked. They describe a gap between known consistency upper and lower bounds rather than an equiconsistency theorem. The assertion quantifies over all finite projective levels at once. Results for a fixed level, for analytic sets, or for lightface definability do not automatically settle it. Later open-question notes by Khomskii and set-theory seminar notes were checked for the same Ramsey regularity problem. They continue to record an unresolved consistency-strength gap. The base theory and the precise meaning of projective parameters affect consistency calibrations. This record fixes \\(\\mathsf{ZFC}\\) and boldface projective subsets of the standard Polish space \\([\\omega]^\\omega\\). Trap: Solovay-model results about all sets of reals often use \\(\\mathsf{ZF}+\\mathsf{DC}\\) without full choice. They cannot be transferred to the \\(\\mathsf{ZFC}\\) statement without a separate argument.\n\nA complete resolution must satisfy: Prove both relative-consistency directions between \\(\\mathsf{ZFC}\\) plus projective Ramsey regularity and a precisely stated standard large-cardinal theory, thereby fixing the exact consistency strength. For the named subquestion, prove equiconsistency with one inaccessible cardinal or give strictly sharper matching upper and lower bounds that refute that calibration.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/76280/exact-consistency-strength-of-all-projective-sets-are-ramsey",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/76280/exact-consistency-strength-of-all-projective-sets-are-ramsey",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
    {
      "slug": "R1355",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
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    {
      "slug": "R1716",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
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    {
      "slug": "projective-sets-ramsey-consistency-strength",
      "title": "projective sets ramsey consistency strength",
      "object_type": "problem",
      "relation": "recorded_for",
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}