{
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  "source": "vibemath",
  "problem_id": "vibemathed:kaul-mudrock-unlabeled-donner-conjecture",
  "title": "Kaul-Mudrock Conjecture on the Unlabeled List Color Function",
  "canonical_statement": "Donner proved in 1992 that the list color function $P_\\ell(G,k)$ equals the chromatic polynomial $P(G,k)$ once $k$ is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result whenever all of its components do.",
  "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:kaul-mudrock-unlabeled-donner-conjecture:event",
      "problem_id": "vibemathed:kaul-mudrock-unlabeled-donner-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-18T00:00:00.000Z",
      "title": "arXiv:2607.16810 - The unlabeled list color function of disconnected 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",
      "attempt_ids": [
        "vibemathed:kaul-mudrock-unlabeled-donner-conjecture:attempt"
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  "attempts": [
    {
      "id": "vibemathed:kaul-mudrock-unlabeled-donner-conjecture:attempt",
      "problem_id": "vibemathed:kaul-mudrock-unlabeled-donner-conjecture",
      "model": "ChatGPT 5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-07-18T00:00:00.000Z",
      "human_collaborators": [
        "Hemanshu Kaul",
        "Jeffrey A. Mudrock",
        "Armin Straub",
        "W. T. Gowers"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "As part of an experiment, Gowers asked ChatGPT 5.5 Pro to find an open combinatorics problem it considered approachable and try to solve it. The model picked this conjecture unprompted and produced an affirmative proof, which Gowers passed to the authors and they confirmed correct. The authors had reached the same theorem independently by a shifting argument, so the model does not hold priority, but its proof is genuinely different (a shadow inequality in the style of the local LYM inequality) and appears in the appendix. Its output also revealed the general form of Corollary 1.6, which the authors had previously established only for complete graphs. Asked to settle the underlying question for all graphs, the model could not.",
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          "license": null
        },
        {
          "label": "VibeMathed: Kaul-Mudrock Conjecture on the Unlabeled List Color Function",
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          "kind": "source-assertion",
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      "solution_event_id": "vibemathed:kaul-mudrock-unlabeled-donner-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The authors state that they checked the AI-produced proof and found it correct, and they include a cleaned-up version of it in Appendix A. arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Kaul-Mudrock Conjecture on the Unlabeled List Color Function",
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          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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        {
          "label": "arXiv:2607.16810 - The unlabeled list color function of disconnected graphs",
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      "url": "https://arxiv.org/abs/2607.16810",
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      "license": null
    },
    {
      "label": "VibeMathed: Kaul-Mudrock Conjecture on the Unlabeled List Color Function",
      "url": "https://vibemathed.com/problem/kaul-mudrock-unlabeled-donner-conjecture",
      "kind": "source-assertion",
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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."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}