{
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  "problem_id": "vibemathed:sivaraman-perfect-divisibility-question-p17",
  "title": "Sivaraman's Perfect-Divisibility Characterization Question",
  "canonical_statement": "Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\\chi(H) \\le \\binom{\\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.",
  "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",
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    {
      "id": "vibemathed:sivaraman-perfect-divisibility-question-p17:event",
      "problem_id": "vibemathed:sivaraman-perfect-divisibility-question-p17",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "Perfect Divisibility, Linear Divisibility and Chair-Free Graphs",
      "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",
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      "id": "vibemathed:sivaraman-perfect-divisibility-question-p17:attempt",
      "problem_id": "vibemathed:sivaraman-perfect-divisibility-question-p17",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Zhiyu Wang",
        "Weihao Xia"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement states that the Paley graph P(17) - the counterexample witness - was identified during exploratory use of ChatGPT and subsequently verified by the authors, with ChatGPT also used for language polishing and generating the verification code. All arguments and computations were checked by the authors.",
      "failed_routes": [],
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        {
          "label": "Perfect Divisibility, Linear Divisibility and Chair-Free Graphs",
          "url": "https://arxiv.org/abs/2608.14519",
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          "license": null
        },
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          "label": "VibeMathed: Sivaraman's Perfect-Divisibility Characterization Question",
          "url": "https://vibemathed.com/problem/sivaraman-perfect-divisibility-question-p17",
          "kind": "source-assertion",
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      "solution_event_id": "vibemathed:sivaraman-perfect-divisibility-question-p17: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:2608.14519, Wang-Xia): the question is stated as Question 1 with attribution to Sivaraman by personal communication, the negative answer via P(17) is Theorem-level, and the acknowledgement is verbatim as quoted. The P(17) computation was not re-run here. The paper's larger content - perfect and linear divisibility of chair-free graphs - is human work outside this entry's claim. Days old, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Sivaraman's Perfect-Divisibility Characterization Question",
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          "label": "Perfect Divisibility, Linear Divisibility and Chair-Free Graphs",
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  "sources": [
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      "label": "Perfect Divisibility, Linear Divisibility and Chair-Free Graphs",
      "url": "https://arxiv.org/abs/2608.14519",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Sivaraman's Perfect-Divisibility Characterization Question",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}