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 C (([n])(k)), they must satisfy min{vertical bar A vertical bar, vertical bar B vertical bar <= 1/2 ((n-1)(k-1))? We give an affirmative answer for n >= 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 >= 3k, the maximum diversity of an intersecting family F subset of (([n])(k)) ( )is equal to ((n-3)(k-2)) We are able to find a construction beating the conjectured bound for n slightly larger than 3k, which also disproves a conjecture of Kupayskii. (C) 2018 Elsevier Ltd. All rights reserved.