Title: Counting intersecting and pairs of cross-intersecting families,
Abstract: A family of subsets of $\{1,\ldots,n\}$ is called {\it 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 $\{1,\ldots, n\}$. In this paper we study the following
problem: how many intersecting families of $k$-subsets of $\{1,\ldots, n\}$ are
there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we
determine this quantity asymptotically for $n\ge 2k+2+2\sqrt{k\log k}$ and
$k\to \infty$. Moreover, under the same assumptions we also determine
asymptotically the number of {\it 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.