{
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  "source": "vibemath",
  "problem_id": "vibemathed:tournament-isomorphism-bounded-vc-dimension",
  "title": "Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension",
  "canonical_statement": "Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \\log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number is also polynomial-time decidable.",
  "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:tournament-isomorphism-bounded-vc-dimension:event",
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      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "Isomorphism of tournaments with bounded VC dimension",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:tournament-isomorphism-bounded-vc-dimension:attempt"
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  "attempts": [
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      "id": "vibemathed:tournament-isomorphism-bounded-vc-dimension:attempt",
      "problem_id": "vibemathed:tournament-isomorphism-bounded-vc-dimension",
      "model": "Claude Sonnet 5",
      "provider": "Anthropic",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Simon Rassmann",
        "Pascal Schweitzer"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The statement of AI use names two specific steps: the proof of one preliminary lemma (the tournament VC-dimension lemma) was provided by Claude Sonnet 5, and the proof of a second bound was simplified by it. Named model-contributed lemmas inside a human-led argument.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Isomorphism of tournaments with bounded VC dimension",
          "url": "https://arxiv.org/abs/2608.14486",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension",
          "url": "https://vibemathed.com/problem/tournament-isomorphism-bounded-vc-dimension",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:tournament-isomorphism-bounded-vc-dimension:verification",
      "problem_id": "vibemathed:tournament-isomorphism-bounded-vc-dimension",
      "solution_event_id": "vibemathed:tournament-isomorphism-bounded-vc-dimension:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14486, Rassmann-Schweitzer): the statement of AI use is verbatim, with the two lemmas identified by reference, and the abstract states the Neuen-Grohe attribution. The algorithm was not checked here. Days-old preprint, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension",
          "url": "https://vibemathed.com/problem/tournament-isomorphism-bounded-vc-dimension",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Isomorphism of tournaments with bounded VC dimension",
          "url": "https://arxiv.org/abs/2608.14486",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "Isomorphism of tournaments with bounded VC dimension",
      "url": "https://arxiv.org/abs/2608.14486",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension",
      "url": "https://vibemathed.com/problem/tournament-isomorphism-bounded-vc-dimension",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:tournament-isomorphism-bounded-vc-dimension:signal:watch_only",
      "problem_id": "vibemathed:tournament-isomorphism-bounded-vc-dimension",
      "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"
}