Source authenticated

Erdős Problem #670: Diameter with Separated Distances

Erdős asked whether every $n$-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least $(1+o(1))n^2$. Disproved: an explicit high-dimensional construction beats the conjectured constant.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Apr 16, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: preprint. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: Erdős Problem #670: Diameter with Separated Distances · Erdős #670 · Problem 670

Confidence: Not scored

Registry verification: lean checked · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

Boon Suan Ho
human · human collaborator

GPT-5.4 Pro
model · ai model contributor · OpenAI

Harmonic Aristotle
model · ai model contributor · Harmonic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Erdős Problem #670: Diameter with Separated Distances

specialist registry · passed

The proof is formalized in Lean 4 by Harmonic Aristotle; the formalization is public. Tier: the formalization is by Harmonic Aristotle with author verification only - and as of August 2026, erdosproblems.com still lists #670 as OPEN, so the canonical tracker has not yet accepted the disproof.

Source ↗

Erdős Problem #670: Diameter with Separated Distances

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean 4 formalization by Harmonic Aristotle

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean 4 formalization by Harmonic Aristotle

lean artifact · pending