  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 $n\geq 2k +
c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars.
Improving their result, we show that the same conclusion holds for $n\geq 2k+
100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and
the Das-Tran removal lemma.
