Erdős Problem #654
Erdős problem #654 · erdosproblems.com/654
If planar points have no four concyclic, must some point determine distinct distances? Failing that, can one always force more than ?
- Result
- Disproved(see note)
- Status
- Variant only
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Combinatorial Geometry
- Posed by
- —
- Year posed
- 1987
- Years open
- 39y
- Solved
- 2026-02
- Model
- Aletheia (Gemini Deep Think)
- Vendor
- Google DeepMind
- Collaborators
- —
- Verification
- Independently expert-verified
- Publication
- Announced
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
the strongest form is disproved via configurations where every point sees at most about 3n/4 distinct distances; the weaker improvement remains open
Verification
Expert-reviewed within the Aletheia project, with public report and transcripts; no journal review.