18 Borsuk’s conjecture
Let \(q = p^m\) be a prime power, \(n := 4q - 2\), and \(d := \binom {n}{2} = (2q - 1)(4q - 3)\). Then there is a set \(S \subseteq \{ +1, -1\} ^d\) of \(2^{n-2}\) points in \(\mathbb {R}^d\) such that every partition of \(S\), whose parts have smaller diameter than \(S\), has at least
\[ \frac{2^{n-2}}{\sum _{i=0}^{q-2} \binom {n-1}{i}} \]
parts. For \(q = 9\) this implies that the Borsuk conjecture is false in dimension \(d = 561\). Furthermore, \(f(d) {\gt} (1.2)\sqrt{d}\) holds for all large enough \(d\).
Proof
TODO