Two extremal problems on intersecting families

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $$\mathcal{A}$, $\mathcal{B}$ \subset \binom{[n]}{k}$,theymustsatisfy$\min{|$\mathcal{A}$|, |$\mathcal{B}$|} \le \frac{1}{2} \binom{n-1}{k-1}$? We give an affirmative answer for $n\ge 2k^2$, and construct families showing that this range is essentially the best one could hope for, up to a constant factor. The second problem is a conjecture of Frankl. It states that for $n\ge 3k$, the maximum diversity of an intersecting family $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ is equal to $\left(\genfrac{}{}{0px}{}{n-3}{k-2}\right)$. We are able to find a construction beating the conjectured bound for $n$ slightly larger than $3k$, which also disproves a conjecture of Kupavskii.