{
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  "source": "vibemath",
  "problem_id": "vibemathed:erdos-graham-unit-fraction-averages",
  "title": "Erdos-Graham Question on Averages of Unit Fractions",
  "canonical_statement": "Erdos and Graham asked whether a positive-density subset of $\\{1,\\ldots,N\\}$ can avoid having any two distinct elements $a,b$ whose unit fractions average to a unit fraction. It can: there is a constant $c>0$ such that for all large $N$ some $A \\subseteq \\{1,\\ldots,N\\}$ of size $> cN$ has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.",
  "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:erdos-graham-unit-fraction-averages:event",
      "problem_id": "vibemathed:erdos-graham-unit-fraction-averages",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-16T00:00:00.000Z",
      "title": "arXiv:2607.15419 - Sets of unit fractions without two members whose average is a unit fraction",
      "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.",
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        "vibemathed:erdos-graham-unit-fraction-averages:attempt"
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      "provider": "OpenAI",
      "attempted_at": "2026-07-16T00:00:00.000Z",
      "human_collaborators": [
        "Will Sawin"
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      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author gives a narrative rather than a blanket acknowledgement. Starting from a computation of Stijn Cambie, he asked ChatGPT to look for patterns in Cambie's extremal set that might suggest a generalization; it observed that in a pair with a given ratio the larger element is usually absent unless the smaller is absent for other reasons, and described a change of variables. Dropping the hedges in that observation gives the set the paper analyzes, which turns out to be essentially where Hooley's function takes its minimum value. The model was also used for reference search and proofreading.",
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          "kind": "primary-mathematical-source",
          "license": null
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        {
          "label": "VibeMathed: Erdos-Graham Question on Averages of Unit Fractions",
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          "kind": "source-assertion",
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      "verified_at": null,
      "note": "Single-author arXiv preprint; not yet peer-reviewed.",
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          "label": "VibeMathed: Erdos-Graham Question on Averages of Unit Fractions",
          "url": "https://vibemathed.com/problem/erdos-graham-unit-fraction-averages",
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          "label": "arXiv:2607.15419 - Sets of unit fractions without two members whose average is a unit fraction",
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  "sources": [
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      "label": "arXiv:2607.15419 - Sets of unit fractions without two members whose average is a unit fraction",
      "url": "https://arxiv.org/abs/2607.15419",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdos-Graham Question on Averages of Unit Fractions",
      "url": "https://vibemathed.com/problem/erdos-graham-unit-fraction-averages",
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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"
}