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 independently showed that for $ngeq 2k + csqrt{kln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $ngeq 2k+ 100ln k$. Our proof uses, among others, Sapozhenkos graph container lemma and the Das-Tran removal lemma.