{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:unit-distance-independence-ratio",
  "title": "Erdos's Question on the Independence Ratio of Unit-Distance Graphs",
  "canonical_statement": "Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below $1/4$. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen two-vertex augmentation.",
  "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:unit-distance-independence-ratio:event",
      "problem_id": "vibemathed:unit-distance-independence-ratio",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-06-26T00:00:00.000Z",
      "title": "arXiv:2606.28157 - A unit-distance graph in the plane with independence ratio below 1/4",
      "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_assisted",
      "attempt_ids": [
        "vibemathed:unit-distance-independence-ratio:attempt"
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      "method_family_ids": [
        "vibemathed:unit-distance-independence-ratio:method"
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      "source_assertion_ids": [
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  "attempts": [
    {
      "id": "vibemathed:unit-distance-independence-ratio:attempt",
      "problem_id": "vibemathed:unit-distance-independence-ratio",
      "model": "ChatGPT, Codex (GPT-5.5)",
      "provider": "OpenAI",
      "attempted_at": "2026-06-26T00:00:00.000Z",
      "human_collaborators": [
        "Akos Ducz",
        "Daniel Varga"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The models were used for software development including code drafting and debugging, plus improvement and formatting suggestions. The search that produces the graph is computational, so the tooling is load-bearing even though no mathematical step is attributed.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2606.28157 - A unit-distance graph in the plane with independence ratio below 1/4",
          "url": "https://arxiv.org/abs/2606.28157",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdos's Question on the Independence Ratio of Unit-Distance Graphs",
          "url": "https://vibemathed.com/problem/unit-distance-independence-ratio",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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  ],
  "verifications": [
    {
      "id": "vibemathed:unit-distance-independence-ratio:verification",
      "problem_id": "vibemathed:unit-distance-independence-ratio",
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      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The result is a specific finite graph, so it is checkable by computation. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Erdos's Question on the Independence Ratio of Unit-Distance Graphs",
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          "label": "arXiv:2606.28157 - A unit-distance graph in the plane with independence ratio below 1/4",
          "url": "https://arxiv.org/abs/2606.28157",
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  "sources": [
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      "label": "arXiv:2606.28157 - A unit-distance graph in the plane with independence ratio below 1/4",
      "url": "https://arxiv.org/abs/2606.28157",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdos's Question on the Independence Ratio of Unit-Distance Graphs",
      "url": "https://vibemathed.com/problem/unit-distance-independence-ratio",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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      "id": "vibemathed:unit-distance-independence-ratio:signal:watch_only",
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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"
}