Erdős problem 505

Erdős Problem 505 (disproved). Borsuk's conjecture is false for sufficiently large $n$: there exists a dimension $n$ and a bounded set $S \subseteq \mathbb{R}^n$ with positive diameter such that $S$ cannot be covered by $n + 1$ subsets each of diameter strictly less than $\operatorname{diam}(S)$.

Erdős [Er44] suspected this. Disproved by Kahn–Kalai [KK93] for $n \geq 2015$. Currently known to be false for $n \geq 64$. A formal proof was formalised by Boris Alexeev using Aristotle.