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

Theorem (Frankl–Wilson, 1981). Let $k$ be a prime power and $n = 4k$. Let $\mathcal{K}$ be a family of $n/2$-subsets of $\{1, \ldots, n\}$ such that no two sets in $\mathcal{K}$ intersect in exactly $n/4$ elements. Then $|\mathcal K| \le 2\binom{n-1}{n/4-1}$.

The striking feature of this bound is its shape: the total number of $n/2$-subsets of an $n$-set is $\binom{n}{n/2}$, exponentially larger (relative to the dimension) than the bound $2\binom{n-1}{n/4-1}$, which only counts subsets of size roughly $n/4$ from a slightly smaller ground set. So forbidding a single intersection size — an extremely mild-sounding restriction — forces a family to be a vanishing fraction of all possible sets, a gap that grows without bound as $n \to \infty$. A closely related result of Frankl and Vojtěch Rödl, answering a question of Erdős, gives a similar but weaker bound $|\mathcal K| \le (1.99)^n$ under the same hypothesis for any $n$ divisible by four, without requiring $k=n/4$ to be a prime power.