Erdős Problem #953

Precise statement

What is the largest possible measure of a subset of a radius-$R$ disk in $\mathbb{R}^2$ containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives $M(R) \ll R^{1/2}$; with Sárközy's lower construction, $M(R) = R^{1/2 + o(1)}$.

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

GPT-5.5 Pro

The source does not provide a sufficiently specific role description.

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

Verification boundary

unreviewed

Public proof; expert digestion ongoing.

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

Timeline

1. 2026-04-27

   erdosproblems.com/953

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

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

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

• The source status is candidate and must not be represented as solved.
• 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.