Intersecting families of sets are typically trivial

A family of subsets of [n] 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 >= 2k + c root k ln k, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for n >= 2k + 100 ln k. Our proof uses, among others, the graph container method and the Das-Tran removal lemma.(c) 2023 Elsevier Inc. All rights reserved.