{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture",
  "title": "Erdős Problem #548: the Erdős–Sós conjecture",
  "canonical_statement": "Let $n\\ge k+1$. Every graph $G$ on $n$ vertices with\n$$\n|E(G)|\\ge \\frac{k-1}{2}n+1\n$$\ncontains every tree on $k+1$ vertices as a not necessarily induced subgraph.",
  "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-548-erdos-sos-conjecture:event",
      "problem_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-26T00:00:00.000Z",
      "title": "Github",
      "summary": "For every $n,k$ with $k+1\\le n$, every simple graph $G$ on $n$ vertices satisfying\n$$\n|E(G)|\\ge \\frac{k-1}{2}n+1\n$$\ncontains every tree on $k+1$ vertices.\n\nThe proof counts pairs $(\\pi,j)$ where $\\pi=(v_1,\\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly\n$$\n2|E(G)|(n-1)!\n$$\nsuch pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus\n$$\n(k-1)n!.\n$$\nIf the target tree is absent, the rooted-copy term vanishes and one obtains\n$$\n2|E(G)|\\le (k-1)n,\n$$\ncontradicting the density hypothesis.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-problem-548-erdos-sos-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-problem-548-erdos-sos-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-problem-548-erdos-sos-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-problem-548-erdos-sos-conjecture:attempt",
      "problem_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture",
      "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 the Lean formalization in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. The proof uses a permutation-word counting argument: it counts ordered host-vertex configurations whose first vertex is adjacent to a later vertex, then uses two reversible word operations and induction on the target tree. If the tree is absent, the counting inequality yields\n$$\n2|E(G)|\\le (k-1)n,\n$$\ncontradicting the assumed edge density.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "erdosproblems.com/548: status and Thomas Bloom's proof exposition",
          "url": "https://www.erdosproblems.com/548",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Challenge.lean: the compared statement",
          "url": "https://github.com/tadamcz/erdos548/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Solution.lean and the Lean development",
          "url": "https://github.com/tadamcz/erdos548/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/erdos548",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #548: the Erdős–Sós conjecture",
          "url": "https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 363,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-problem-548-erdos-sos-conjecture:verification",
      "problem_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture",
      "solution_event_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture: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/erdos548 at 3766491: 1,311 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 FrontierMath Erdős benchmark and reviewed by Thomas Bloom, and Challenge.lean is copied from that file; it follows erdosproblems.com's phrasing, which differs from the classical one by requiring one extra edge when (t-2)n is odd, a parity margin the repository's README discusses and the internal counting lemma closes. 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 #548: the Erdős–Sós conjecture",
          "url": "https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos548",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/erdos548",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #548: the Erdős–Sós conjecture",
      "url": "https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-problem-548-erdos-sos-conjecture:signal:replay_ready",
      "problem_id": "vibemathed:erdos-problem-548-erdos-sos-conjecture",
      "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"
}