{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-848",
  "title": "Erdős Problem #848",
  "canonical_statement": "Is the maximum size of a set $A\\subseteq \\{1,\\ldots,N\\}$ such that $ab+1$ is never squarefree (for all $a,b\\in A$) achieved by taking those $n\\equiv 7\\pmod{25}$? Resolved for all sufficiently large $N$: any near-maximal $A$ is contained in $\\{n\\equiv 7\\pmod{25}\\}$ or $\\{n\\equiv 18\\pmod{25}\\}$, leaving only a finite check.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:erdos-848:event",
      "problem_id": "vibemathed:erdos-848",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2025-11-20T00:00:00.000Z",
      "title": "erdosproblems.com/848",
      "summary": "Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-848:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-848:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-848:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-848:attempt",
      "problem_id": "vibemathed:erdos-848",
      "model": "GPT-5",
      "provider": "OpenAI",
      "attempted_at": "2025-11-20T00:00:00.000Z",
      "human_collaborators": [
        "Mehtaab Sawhney",
        "Mark Sellke"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Sawhney's note resolving the problem cites a ChatGPT (GPT-5) session in the provenance of the key lemma, and Tao's AI-contributions ledger records the solve as GPT-5 working with Sawhney and Sellke (October-November 2025).",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Sawhney's note resolving the problem",
          "url": "https://www.math.columbia.edu/~msawhney/Problem_848.pdf",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/848",
          "url": "https://www.erdosproblems.com/848",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #848",
          "url": "https://vibemathed.com/problem/erdos-848",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-848:verification",
      "problem_id": "vibemathed:erdos-848",
      "solution_event_id": "vibemathed:erdos-848:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "erdosproblems.com marks the problem resolved up to a finite check and links Sawhney's note; no formal artifact and no journal review.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #848",
          "url": "https://vibemathed.com/problem/erdos-848",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/848",
          "url": "https://www.erdosproblems.com/848",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/848",
      "url": "https://www.erdosproblems.com/848",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #848",
      "url": "https://vibemathed.com/problem/erdos-848",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-848:signal:watch_only",
      "problem_id": "vibemathed:erdos-848",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "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"
}