Borsuk Conjecture lowest-ever counterexample (N=63)
We exhibit a set of 321 points in R^63 that cannot be partitioned into 64 subsets of smaller diameter, so Borsuk’s conjecture fails in dimension 63. The previous smallest counterexample was in dimension 64 (Jenrich, 2014), and the conjecture was open for 4 <= n <= 63. The construction adds a single point to the 320-point rank-63 subconfiguration of Bondarenko’s two-distance set, whose counting bound has been stuck at exactly 64 parts. The added point is not a vertex of the underlying strongly regular graph, so the resulting set is a three-distance set; this is precisely why the example was not reachable inside the two-distance framework in which all previous work took place.
- Result
- Disproved
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Discrete geometry
- Posed by
- Karol Borsuk
- Year posed
- 1933
- Years open
- 93y
- Solved
- 2026-08-11
- Model
- Claude Fable 5 and Opus 5
- Vendor
- Anthropic
- Collaborators
- Nicholas Konz
- Verification
- Site-confirmed
- Publication
- Announced
- Significance
- —
- Disclosed cost
- —
- Wikipedia
- No dedicated article
Verification
Reproduced by this site on 12 August 2026. We ran the author's stand-alone verifier against the published 321x63 coordinate file and independently confirmed all of it: affine dimension exactly 63, a three-value squared-distance spectrum of (53-sqrt(222))/156, 1/4 and 1/3, and independence number 5 for the diameter graph by Bron-Kerbosch over all 321 points. That forces ceil(321/5) = 65 parts where Borsuk allows 64. The 1/4-to-1/3 gap is far wider than any floating-point tolerance, so the computed diameter graph is the exact one. Claude's own checks were an exact rational and algebraic certificate over Q(sqrt(222)), two independently written max-clique implementations, an ILP solver and CP-SAT. No independent expert has reviewed the write-up, and it is not peer-reviewed or on arXiv.
Source
- OtherNick Konz
Submitted by WildWalrus807 on