Intersecting families of sets are usually trivial for $n\ge 2k+3$

Abstract

A family of subsets of $[n]$ is called intersecting if it contains no pair of disjoint sets. It is called trivial if all its members contain a common element. Frankl and Kupavskii, and independently Balogh, Das, Liu, Sharifzadeh, and Tran, proved that there is a constant $c>0$ such that, whenever $n \geq 2k+2+c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are trivial. Balogh, Garcia, Li, and Wagner later improved this range to $n \geq 2k+100\ln k$. In this paper, we prove that the same conclusion holds for every $n\geq 2k+3$. This verifies the conjectured conclusion of Balogh, Garcia, Li, and Wagner throughout this range.