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Borsuk Conjecture lowest-ever counterexample (N=63)

Borsuk's conjecture asked whether every bounded set in Rn\mathbb{R}^n can be partitioned into n+1n+1 subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in R63\mathbb{R}^{63} whose smaller-diameter subsets have at most 5 points, so at least ⌈321/5⌉=65>64\lceil 321/5\rceil = 65 > 64 parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for 4≤n≤624 \le n \le 62. The construction modifies Bondarenko's G2(4)G_2(4) two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.

Result
Disproved(see note)
Status
Partial result
AI contribution
AI-assisted
Method
Construction
Field
Discrete geometry
Posed by
Karol Borsuk
Year posed
1933
Years open
93y
Solved
2026-05-26
Model
GPT-5.5 Pro
Vendor
OpenAI
Collaborators
Max Grinsztajn
Verification
Site-confirmed
Publication
Preprint
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.

What the AI did

For the first solve, Grinsztajn's README states: "The construction and proof were obtained with assistance from GPT-5.5 Pro", with a dedicated "Disclose GPT assistance" commit; no finer division of labour is given, so the tier is the floor for an unspecific disclosure. The independent August 2026 rediscovery by Nicholas Konz with Claude (Fable 5 and Opus 5) carries a much fuller disclosure - Claude produced the counterexample and an exact certificate over Q(222)\mathbb{Q}(\sqrt{222}) - and would rate ai-discovered on its own, but the entry's tier follows the solve it records, which is the first one.

Verification

Both derivations reproduced by this site on 12 August 2026, independently of each other. For the first solve (Grinsztajn, May 2026): the repository's exact verifier - pure Python integer arithmetic over F16, read before running - was executed locally and passes all checks: it rebuilds the G2(4) strongly regular graph with parameters (416,100,36,20), the B1/B2/B3/C partition and degree data behind the dimension drop, and the clique obstructions forcing every smaller-diameter subset to size at most 5. The repo's GitHub creation date of 2026-05-26 is not forgeable after the fact, and Terence Tao's optimization-problems ledger (constant 28a) independently credits the 63 bound to Grinsztajn, citing this repository. For the August rediscovery (Konz + Claude): we ran the author's stand-alone verifier against the published 321x63 coordinate file and confirmed affine dimension exactly 63, the squared-distance spectrum (53-sqrt(222))/156, 1/4 and 1/3, and independence number 5 for the diameter graph by Bron-Kerbosch, forcing ceil(321/5) = 65 parts where Borsuk allows 64; the distance-class gap is far wider than any float tolerance. Neither write-up is peer-reviewed; neither is on arXiv.

Sources

FrontierStep on Borsuk counterexample dimension · 63 · current best

Submitted by WildWalrus807 on

Changelog6 changes
  • Rasmus Lindahlchanged Links from “1 link repeating the primary source” to “removed / shortened to satisfy the link rules”
  • Rasmus Lindahlset Significance note to “Borsuk's conjecture is a famous named problem with a Wikipedia article and a 90-year histo…”, also Model maker, Age note, Solve date, AI contribution, Model, Verification note, Collaborators, Source, Statement, Source name, Publication, What was actually shown, Significance, AI role
  • Rasmus Lindahlchanged Verification note from “Claude verified the counterexample in four independent ways: an exact rational/algebraic-a…” to “Reproduced by this site on 12 August 2026. We ran the author's stand-alone verifier agains…”
  • Rasmus Lindahlapproved this entry
  • Rasmus Lindahlset Collaborators to “Nicholas Konz”
  • WildWalrus807submitted this entry

Discussion1

Rasmus Lindahl12 Aug 2026, 20:16 UTC

Priority correction: this result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance on 26 May 2026 (github.com/maaxgrin/borsuk-63-counterexample), roughly ten weeks before the Konz + Claude derivation this entry originally recorded. The submitter surfaced the earlier work themselves after publication. This site verified the priority claim (unforgeable repo creation date, exact verifier passes locally, and Tao's optimization-problems ledger already credits the bound to Grinsztajn) and re-centred the entry on the first solve. The Konz + Claude work remains in the entry as a verified independent rediscovery with a different derivation and a fuller AI disclosure.

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