Erdős's Planar Unit Distance Conjecture
Erdős problem #90 · erdosproblems.com/90
Conjectured upper bound on how many pairs among points in the plane can be exactly one unit apart.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Combinatorial Geometry
- Posed by
- Paul Erdős
- Year posed
- 1946
- Years open
- 80y
- Solved
- 2026-05
- Model
- OpenAI frontier model (specific version not disclosed)
- Vendor
- OpenAI
- Collaborators
- Noga Alon, Thomas Bloom, Timothy Gowers, Daniel Litt, Will Sawin, Jacob Tsimerman, Melanie Matchett Wood
- Verification
- Independently expert-verified
- Publication
- Announced
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- 10 languages
What the AI did
Model-assisted construction of a point configuration with more than unit-distance pairs, beating the conjectured bound.
Verification
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.