{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-1186",
  "title": "Erdős Problem #1186",
  "canonical_statement": "What is the minimum asymptotic density $\\delta_k$ of monochromatic $k$-term arithmetic progressions in every two-colouring of $\\{1, \\dots, n\\}$? The exact certificate gives $\\delta_3 = 117/2192$, matching the known 548-bead colouring.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:erdos-1186:event",
      "problem_id": "vibemathed:erdos-1186",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "erdosproblems.com/1186",
      "summary": "exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-1186:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-1186:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-1186:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-1186:attempt",
      "problem_id": "vibemathed:erdos-1186",
      "model": "GPT-5.6 Star Fleet (Claude Fable 5 referee)",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-13T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": null,
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "erdosproblems.com/1186",
          "url": "https://www.erdosproblems.com/1186",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #1186",
          "url": "https://vibemathed.com/problem/erdos-1186",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-1186:verification",
      "problem_id": "vibemathed:erdos-1186",
      "solution_event_id": "vibemathed:erdos-1186:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Dual exact checkers plus a partial Lean formalization; community status pending.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #1186",
          "url": "https://vibemathed.com/problem/erdos-1186",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/1186",
          "url": "https://www.erdosproblems.com/1186",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/1186",
      "url": "https://www.erdosproblems.com/1186",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #1186",
      "url": "https://vibemathed.com/problem/erdos-1186",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-1186:signal:recent_partial_result",
      "problem_id": "vibemathed:erdos-1186",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}