Intersecting families of sets are typically trivial

Abstract

A family of subsets of $\left[n\right]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for $n\ge 2k+c\sqrt{k\ln k}$, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\ge 2k+100\ln k$. Our proof uses, among others, the graph container method and the Das–Tran removal lemma.