Erdős's Planar Unit Distance Conjecture

Precise statement

Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.

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

OpenAI frontier model (specific version not disclosed)

Model-assisted construction of a point configuration with more than $n^{1.014}$ unit-distance pairs, beating the conjectured bound.

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

Verification boundary

independent expert verified

The counterexample was generated by an OpenAI model; the human-verified version was written up by Noga Alon, Thomas Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman and coauthors, who call it a short, digested, human-verified version of the construction. They attribute the crucial ideas, in retrospect, to Ellenberg-Venkatesh, Golod-Shafarevich and Hajir-Maire-Ramakrishna.

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

Timeline

1. 2026-05-01

   arXiv:2605.20695 - Remarks on the disproof of the unit distance conjecture

   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.