{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:symmetrically-colored-k-ap-bound",
  "title": "Improved Bound for Colorings Without Symmetrically Colored k-APs",
  "canonical_statement": "Deng, Tidor and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no symmetrically coloured 4-term arithmetic progression, giving an $O(N^{\\log_{22}3})$ coloring. The paper gives an $O_k(N^{4/k^2})$ coloring of $[N]$ avoiding symmetrically coloured $k$-APs for every even $k\\ge4$, improving the exponent.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:symmetrically-colored-k-ap-bound:event",
      "problem_id": "vibemathed:symmetrically-colored-k-ap-bound",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:symmetrically-colored-k-ap-bound:attempt"
      ],
      "method_family_ids": [
        "vibemathed:symmetrically-colored-k-ap-bound:method"
      ],
      "source_assertion_ids": [
        "vibemathed:symmetrically-colored-k-ap-bound:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:symmetrically-colored-k-ap-bound:attempt",
      "problem_id": "vibemathed:symmetrically-colored-k-ap-bound",
      "model": "GPT-5",
      "provider": "OpenAI",
      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [
        "Ruizhe Shi",
        "Yiqi Dong"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgements state the key construction idea was suggested during an interaction with GPT-5.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.20752",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Improved Bound for Colorings Without Symmetrically Colored k-APs",
          "url": "https://vibemathed.com/problem/symmetrically-colored-k-ap-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:symmetrically-colored-k-ap-bound:verification",
      "problem_id": "vibemathed:symmetrically-colored-k-ap-bound",
      "solution_event_id": "vibemathed:symmetrically-colored-k-ap-bound:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent review. The model is credited with the key construction idea rather than the surrounding analysis. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: Improved Bound for Colorings Without Symmetrically Colored k-APs",
          "url": "https://vibemathed.com/problem/symmetrically-colored-k-ap-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.20752",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.20752",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Improved Bound for Colorings Without Symmetrically Colored k-APs",
      "url": "https://vibemathed.com/problem/symmetrically-colored-k-ap-bound",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:symmetrically-colored-k-ap-bound:signal:recent_partial_result",
      "problem_id": "vibemathed:symmetrically-colored-k-ap-bound",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}