{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-856",
  "title": "Erdős Problem #856",
  "canonical_statement": "Let $k\\geq 3$ and $f_k(N)$ be the maximum of $\\sum_{n\\in A}\\frac{1}{n}$ over all $A\\subseteq\\{1,\\ldots,N\\}$ containing no $k$ subsets with the same pairwise least common multiple. Estimate $f_k(N)$. The claimed answer: $f_k(N)=(\\log N)^{\\gamma_k+o(1)}$, where $\\gamma_k$ is a weighted generalization of the Tang-Zhang sunflower capacity.",
  "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-856:event",
      "problem_id": "vibemathed:erdos-856",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-04-15T00:00:00.000Z",
      "title": "erdosproblems.com/856",
      "summary": "Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-856:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-856:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-856:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-856:attempt",
      "problem_id": "vibemathed:erdos-856",
      "model": "GPT-5.4 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-04-15T00:00:00.000Z",
      "human_collaborators": [
        "Przemysław Chojecki"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A weighted version of the Tang-Zhang sunflower-capacity argument giving the exact logarithmic exponent was developed with GPT-5.4 Pro, using a mass-transport idea from the forum's discussion of problem #1196.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Write-up",
          "url": "https://www.ulam.ai/research/erdos856-final.pdf",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/856",
          "url": "https://www.erdosproblems.com/856",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #856",
          "url": "https://vibemathed.com/problem/erdos-856",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-856:verification",
      "problem_id": "vibemathed:erdos-856",
      "solution_event_id": "vibemathed:erdos-856:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "An AI screening found no issues and no prior literature with the result; the site's owner unpacked and restated the main claim without checking details, a Lean formalization attempt hit missing mathlib prerequisites, and the problem is still listed open.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #856",
          "url": "https://vibemathed.com/problem/erdos-856",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/856",
          "url": "https://www.erdosproblems.com/856",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/856",
      "url": "https://www.erdosproblems.com/856",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #856",
      "url": "https://vibemathed.com/problem/erdos-856",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-856:signal:verify_now",
      "problem_id": "vibemathed:erdos-856",
      "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"
}