Let $\mathcal{A} \subset[n]^{(r)}$ where $r \leqslant n / 2$. Prove that, if $\mathcal{A}$ is 1-intersecting, then $|\mathcal{A}| \leqslant\left(\begin{array}{l}n-1 \\ r-1\end{array}\right)$. State an upper bound on $|\mathcal{A}|$ that is valid if $\mathcal{A}$ is $t$-intersecting and $n$ is large compared to $r$ and $t$.

Let $\mathcal{B} \subset \mathcal{P}([n])$ be maximal 1-intersecting; that is, $\mathcal{B}$ is 1-intersecting but if $\mathcal{B} \subset \mathcal{C} \subset \mathcal{P}([n])$ and $\mathcal{B} \neq \mathcal{C}$ then $\mathcal{C}$ is not 1-intersecting. Show that $|\mathcal{B}|=2^{n-1}$.

Let $\mathcal{B} \subset \mathcal{P}([n])$ be 2 -intersecting. Show that $|\mathcal{B}| \geqslant 2^{n-2}$ is possible. Can the inequality be strict?