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
- Resolved
- 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
- Site-confirmed
- 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
Rebuilt by this site on 6 October 2026. The Lean development added on 4 October (commit 065e5e2; Lean 4.32.1 with mathlib, about 6,760 lines) was built from a clean clone with no warnings, passed a fresh leanchecker kernel replay, and #print axioms gives only propext, Classical.choice and Quot.sound for thm_1_1 and cor_1_2_upper. No sorry, axiom declarations, native_decide or implemented_by; finite steps use decide +kernel. The statement was read: no eight affine lines in R^3 have pairwise distance 1, with lines as affine subspaces of one-dimensional direction and distance the infimum of explicit Euclidean point distances; the corollary covers any n >= 8 and any d > 0. No general-position, non-parallel or catalogue hypothesis. The proof formalises the parallel and coplanar direction lemma, the four-line circuit condition and the five-line obstruction, and replaces Finschi's catalogue with a self-contained kernel-checked enumeration. Separately, on 4 October the Python verifier at 9c37327 was re-run here and passed (135 classes, 1,027 metric certificates). Not formalised: the seven-line construction of Bozoki, Lee and Ronyai, which is published. The statement audit is this site's alone.
Sources
Submitted by SilentIbis759 on