{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:araujo-piga-schacht-tight-hamilton",
  "title": "Araujo-Piga-Schacht Question on Tight Hamilton Cycles",
  "canonical_statement": "Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \\max_{0 \\le x \\le 1}\\min\\{x^3, 1-x\\} \\approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.",
  "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:araujo-piga-schacht-tight-hamilton:event",
      "problem_id": "vibemathed:araujo-piga-schacht-tight-hamilton",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-23T00:00:00.000Z",
      "title": "arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs",
      "summary": "the question is answered negatively and the correct threshold is determined",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:araujo-piga-schacht-tight-hamilton:attempt"
      ],
      "method_family_ids": [
        "vibemathed:araujo-piga-schacht-tight-hamilton:method"
      ],
      "source_assertion_ids": [
        "vibemathed:araujo-piga-schacht-tight-hamilton:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:araujo-piga-schacht-tight-hamilton:attempt",
      "problem_id": "vibemathed:araujo-piga-schacht-tight-hamilton",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-23T00:00:00.000Z",
      "human_collaborators": [
        "Xichao Shu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs",
          "url": "https://arxiv.org/abs/2607.21568",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Araujo-Piga-Schacht Question on Tight Hamilton Cycles",
          "url": "https://vibemathed.com/problem/araujo-piga-schacht-tight-hamilton",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:araujo-piga-schacht-tight-hamilton:verification",
      "problem_id": "vibemathed:araujo-piga-schacht-tight-hamilton",
      "solution_event_id": "vibemathed:araujo-piga-schacht-tight-hamilton:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Araujo-Piga-Schacht Question on Tight Hamilton Cycles",
          "url": "https://vibemathed.com/problem/araujo-piga-schacht-tight-hamilton",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs",
          "url": "https://arxiv.org/abs/2607.21568",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs",
      "url": "https://arxiv.org/abs/2607.21568",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Araujo-Piga-Schacht Question on Tight Hamilton Cycles",
      "url": "https://vibemathed.com/problem/araujo-piga-schacht-tight-hamilton",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:araujo-piga-schacht-tight-hamilton:signal:watch_only",
      "problem_id": "vibemathed:araujo-piga-schacht-tight-hamilton",
      "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"
}