{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-problem-571",
  "title": "Erdős Problem #571: rational exponents for bipartite Turán numbers",
  "canonical_statement": "For every rational $\\alpha\\in[1,2)$, there exists a finite bipartite graph $G$ such that\n$$\n\\operatorname{ex}(n;G)=\\Theta(n^\\alpha).\n$$\nEquivalently, every rational exponent between $1$ and $2$ occurs as the order of growth of the Turán number of a single bipartite graph.",
  "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-problem-571:event",
      "problem_id": "vibemathed:erdos-problem-571",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-26T00:00:00.000Z",
      "title": "Github",
      "summary": "For every rational $\\alpha$ satisfying\n$$\n1\\le\\alpha<2,\n$$\nthe Lean theorem constructs some finite $q$ and a bipartite graph\n$$\nG:\\operatorname{SimpleGraph}(\\operatorname{Fin} q)\n$$\nsuch that\n$$\n\\operatorname{ex}(n;G)=\\Theta(n^\\alpha)\n$$\nas $n\\to\\infty$.\n\nThis resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbidden bipartite graphs, and subsequent work realized many large classes of individual rational exponents. Astra's theorem covers every rational $\\alpha\\in[1,2)$ with one forbidden graph for each exponent.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-problem-571:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-problem-571:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-problem-571:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-problem-571:attempt",
      "problem_id": "vibemathed:erdos-problem-571",
      "model": "GPT-6 Astra (pre-release)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-26T00:00:00.000Z",
      "human_collaborators": [
        "Tom Adamczewski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A pre-release GPT-6 Astra autonomously found the proof and wrote its Lean formalization in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. The roughly 10,000-line development constructs balanced rooted graph models for every rational exponent and proves closure operations that preserve the required extremal-number behavior, including edge subdivision by paths of arbitrary length, adding hubs to the two colour classes, and positive rooted powers. These constructions are combined to realize every rational $\\alpha\\in[1,2)$ by a single finite bipartite graph.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "erdosproblems.com/571: status and Thomas Bloom's proof exposition",
          "url": "https://www.erdosproblems.com/571",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Challenge.lean: the compared statement",
          "url": "https://github.com/tadamcz/erdos571/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Solution.lean and the Lean development",
          "url": "https://github.com/tadamcz/erdos571/blob/main/Solution.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Epoch AI, Announcing FrontierMath Erdős (1 September 2026)",
          "url": "https://epoch.ai/latest/announcing-frontiermath-erdos",
          "kind": "announcement",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos571",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #571: rational exponents for bipartite Turán numbers",
          "url": "https://vibemathed.com/problem/erdos-problem-571",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 617,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-problem-571:verification",
      "problem_id": "vibemathed:erdos-problem-571",
      "solution_event_id": "vibemathed:erdos-problem-571:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos571 at 661cc1d: 10,460 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement was autoformalized for the benchmark and reviewed by Thomas Bloom; it uses mathlib's extremalNumber and IsBipartite and Asymptotics.IsTheta, and the repository's README compares it to the informal statement. One grep hit for the word 'externally' in a docstring is the only match for the risky-feature scan. erdosproblems.com, the field's own record, marks the problem PROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #571: rational exponents for bipartite Turán numbers",
          "url": "https://vibemathed.com/problem/erdos-problem-571",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos571",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/erdos571",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #571: rational exponents for bipartite Turán numbers",
      "url": "https://vibemathed.com/problem/erdos-problem-571",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-problem-571:signal:replay_ready",
      "problem_id": "vibemathed:erdos-problem-571",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}