Worked solution: Kahn–Kalai's disproof of Borsuk's conjecture via the Frankl–Wilson theorem

Step 6 of 7: Explicit counterexamples: from

n = 1{,}325

down to

n = 63

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In plain words

The argument above only guarantees a counterexample once $d$ is astronomically large; turning it into one specific, checkable dimension took real work choosing the smallest prime power $k$ that makes the arithmetic go through. Once the door was open, other mathematicians spent the following decades hunting for cleverer, hand-built or computer-verified configurations that push the failure of Borsuk's conjecture down into dimensions small enough to write out explicitly.

n = 1325 \;(\text{Kahn–Kalai}), \quad n = 65\;(\text{Bondarenko}), \quad n = 64\;(\text{Jenrich–Brouwer}), \quad n=63\;(\text{Ji})

Detailed analysis

Kahn and Kalai's original 1993 argument gave an explicit counterexample only for $n = 1{,}325$ (and all $n > 2{,}014$ ). Subsequent constructions steadily lowered the dimension: Andriy V. Bondarenko (2013) used two-distance sets built from strongly regular graphs to find a counterexample in $\mathbb{R}^{65}$ , Thomas Jenrich and Andries E. Brouwer (2014) found a rigorously verified $352$ -point counterexample in $\mathbb{R}^{64}$ , and a 2026 preprint by Yibo Ji verifies an AI-generated $321$ -point counterexample in $\mathbb{R}^{63}$ , not yet peer-reviewed.

Knowledge used in this step

Convex and discrete geometry