Let $\mathcal A \subseteq X^{(\le \frac n2)}$ be an intersecting antichain. Then
$$\sum_{A \in \mathcal A} \binom{n - 1}{|A| - 1}^{-1} \le 1$$
In particular, $|\mathcal A| \le \binom{n - 1}{r - 1}$.
This is Theorems 1.15 and 1.16 in the lecture notes.
