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 A,B⊂[n]k, they must satisfy min{|A|,|B|}≤12n−1k−1? We give an affirmative answer for n≥2k2, 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≥3k, the maximum diversity of an intersecting family F⊂[n]k is equal to n−3k−2. We are able to find a construction beating the conjectured bound for n slightly larger than 3k, which also disproves a conjecture of Kupavskii.