Counting Intersecting and Pairs of Cross-Intersecting Families

A family of subsets of {1, ... , n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdos, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1, ... , n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1, ... , n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n >= 2k + 2+ 2 root k log k and k -> infinity. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.