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

Borsuk conjectured in 1933 that every bounded set $S \subset \mathbb{R}^n$ can be partitioned into $n+1$ pieces each of diameter strictly less than $d(S) = \max_{x,y \in S} \|x-y\|_2$. The lower bound $f(n) \ge n+1$ is easy (the vertices of a regular simplex, or a ball via the Borsuk–Ulam theorem, both need that many pieces); the conjecture was proved true in dimensions $n=2$ and $n=3$, and for all centrally symmetric or smooth convex bodies in every dimension, which is why it was widely believed.

Before Kahn and Kalai, several authors had suggested that a counterexample, if one existed, would come from combinatorics rather than smooth geometry. In 1965 Ludwig Danzer showed that a finite set of $\{0,1\}$-vectors of a fixed weight cannot be covered by $(1.003)^d$ balls of smaller diameter, a first hint that high-dimensional combinatorial configurations resist small partitions; Paul Erdős and David Larman independently floated the possibility of a genuine counterexample along these lines.