{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-623",
  "title": "Erdős Problem #623",
  "canonical_statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ a function from the finite subsets of $X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite independent $Y\\subseteq X$, i.e. with $f(B)\\not\\in Y$ for all finite $B\\subset Y$? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:erdos-623:event",
      "problem_id": "vibemathed:erdos-623",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-06-04T00:00:00.000Z",
      "title": "erdosproblems.com/623",
      "summary": "Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-623:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-623:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-623:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-623:attempt",
      "problem_id": "vibemathed:erdos-623",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-06-04T00:00:00.000Z",
      "human_collaborators": [
        "Sungchul Lee"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The equivalence to Koepke's free-subset property and the resulting consistency analysis were obtained with the assistance of GPT-5.5 Pro; the author checked the mathematical details.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Paper and LaTeX source",
          "url": "https://github.com/lsngchl/Erdos623",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/623",
          "url": "https://www.erdosproblems.com/623",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #623",
          "url": "https://vibemathed.com/problem/erdos-623",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-623:verification",
      "problem_id": "vibemathed:erdos-623",
      "solution_event_id": "vibemathed:erdos-623:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "An AI screening on the forum found no issues and forum readers concur it would fully resolve the problem in the set-theoretic sense, but there is no independent expert review and erdosproblems.com still lists the problem open.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #623",
          "url": "https://vibemathed.com/problem/erdos-623",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/623",
          "url": "https://www.erdosproblems.com/623",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/623",
      "url": "https://www.erdosproblems.com/623",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #623",
      "url": "https://vibemathed.com/problem/erdos-623",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-623:signal:verify_now",
      "problem_id": "vibemathed:erdos-623",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}