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Borsuk Conjecture lowest-ever counterexample (N=63)

Combinatorics · posed by Karol Borsuk, 1933 · partial

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Statement

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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Max Grinsztajn, using GPT-5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    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.

    Repeated from the source; nothing was checked here.

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