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 Erdős, 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√k log k and k → ∞. 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. MSc classification: 05D05