Discrete geometry·borsuk:dimension
Smallest dimension in which Borsuk's conjecture is known to fail
No independent check recorded yet. A formal artifact, declaration or published object is attached, but no rebuild of it is recorded here.
Fidelity F0: Absent. No formal statement is attached to the result.
Borsuk asked in 1933 whether every bounded set in n-dimensional space can be split into n + 1 parts of smaller diameter. It is true in dimensions 2 and 3 and false in high dimensions; the record is the smallest dimension with a known counterexample.
Record history
6 steps·dimension, lower is betterKahn and Kalai's 1993 counterexample works in dimension 1325; constructions from codes brought the dimension to 298 by 2003, and Bondarenko's two-distance set from the G2(4) graph led to 65 and then 64 in 2013. Several improvements between 1994 and 2002 are not yet drawn here. In May 2026 Max Grinsztajn posted a counterexample in dimension 63, obtained with assistance from GPT-5.5 Pro. In August an arXiv note reported the same dimension from a construction generated by GPT 5.6 Sol, and was withdrawn when its author found the earlier posting.
The note said the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol; its author withdrew it on 14 August on finding the earlier posting.
A 321-point set in which every subset of smaller diameter has at most five points, so at least 65 parts are needed. The repository says the construction and proof were obtained with assistance from GPT-5.5 Pro.
A 352-point subset of Bondarenko's set; published with Andries Brouwer in the Electronic Journal of Combinatorics in 2014.
A 416-point two-distance set from the G2(4) graph; published in Discrete & Computational Geometry in 2014.
New sets with large Borsuk numbers.
The first counterexample; Weißbach later showed the dimension-1325 case needs a corrected argument, which holds.
AI activity
How grades workA 321-point set in dimension 63 that cannot be split into 64 parts of smaller diameter, improving 64.
Reasoning and sources
Autonomy
The repository states the construction and proof were obtained with assistance from GPT-5.5 Pro.
The same dimension, 63, from a construction the author said was generated entirely by ChatGPT; withdrawn on 14 August when he found the earlier posting.
Reasoning and sources
Autonomy
The note stated the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol, with the author verifying them.
Fidelity
How fidelity is gradedF0 no formal statement. Absent. No formal statement is attached to the result.
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qed.bot, “Smallest dimension in which Borsuk's conjecture is known to fail”, https://qed.bot/t/borsuk-dimension, as of 30 Sep 2026.