Erdős Problem #659

Full solution by Benjamin Grayzel with Gemini 3 (13 Jan 2026). 2 AI contributions are recorded in all.

Problem
Erdős problem #659 (geometry, distances); current status: proved (Lean)
Credit
Benjamin Grayzel with Gemini 3 · AI working with humans
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-01-13 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Is there a set of nn points in R2\mathbb{R}^2 such that every subset of 44 points determines at least 33 distances, yet the total number of distinct distances is

≪nlog⁡n?\ll \frac{n}{\sqrt{\log n}}?

Statement quoted from erdosproblems.com/659, maintained by Thomas Bloom. Original source: [Er97e,p.531]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Humans Outcome Literature
13 Jan, 2026 AI working with humans Gemini 3 Benjamin Grayzel 🟢 Full solution —
13 Jan, 2026 Literature search Gemini Deep Research — 🟡 Partial results, upgradeable to full solution, found —
2 Feb, 2026 AI on its own; related literature found afterwards Aletheia — 🟢 Full solution 🟡 Sheffer and Lund (2014); 🟢 Grayzel (2026)

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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