# Counting intersecting and pairs of cross-intersecting families

## TL;DR

This paper asymptotically counts the number of intersecting and cross-intersecting families of k-subsets of an n-set, extending classical extremal combinatorics results to enumeration under certain growth conditions.

## Contribution

It provides the first asymptotic enumeration of intersecting and cross-intersecting families for large n and k, including non-trivial families, improving previous bounds.

## Key findings

- Asymptotic count of intersecting families for large n and k
- Asymptotic enumeration of non-trivial intersecting families
- Results for 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 Erd\H{o}s, 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.