Erdos-Graham Question on Averages of Unit Fractions

Precise 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.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT

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.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Single-author arXiv preprint; not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-16

   arXiv:2607.15419 - Sets of unit fractions without two members whose average is a unit fraction

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• 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.