Maximum number of pairwise equidistant lines in
Is seven the maximum cardinality of a family of affine lines in with a common positive pairwise distance? Equivalently, can eight congruent infinite circular cylinders with pairwise disjoint interiors touch pairwise?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Discrete and computational geometry
- Posed by
- J. E. Littlewood, Some Problems in Real and Complex Analysis (1968), Problem 7, p. 20; the maximum was framed by Bezdek (2005)
- Year posed
- 1968
- Years open
- 58y
- Solved
- 2026-09-30
- Model
- GPT-6 Astra Pro; Claude Fable 5.1
- Vendor
- OpenAI; Anthropic
- Collaborators
- Mingchang Liu
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
It is proved that no eight affine lines in have a common positive pairwise distance. Together with the known construction of seven such lines by Bozóki, Lee and Rónyai, this establishes the exact maximum of seven. The upper bound combines a five-line obstruction from hyperbola secants with exact verification over 135 uniform rank-three oriented-matroid types. It settles the full stated maximum-cardinality question; it does not classify all seven-line configurations or address cylinders of unequal radii.
What the AI did
This work was developed through continuous collaboration between the author and AI models. OpenAI and Anthropic models contributed to proof development, exact computation, literature research, manuscript preparation and review.
Verification
Re-run by this site on 4 October 2026: the companion repository at commit 9c37327, with Finschi's catalogue of rank-three oriented matroids fetched the same day (row hash matched the pinned SHA-256). verify.py passed: 135 classes, 135 exhaustion trees, 153,080 leaves, 1,027 exact metric certificates, about 50 seconds. The independent metric verifier covered all 1,027 pairs, and the input tests passed. Not checked here, and not machine-checked anywhere: the analytic lemmas of the manuscript (the five-line hyperbola-secant obstruction and the four-line circuit condition) and the completeness of the Finschi-Fukuda catalogue, which is published. No Lean formalisation. The manuscript names no model; the model names come from the submission.
Sources
Submitted by SilentIbis759 on