Intersecting families with covering number $3$

A covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for n≥ n₀(k) the largest intersecting family of k-element subsets of $[n]$ with covering number $3$. In this paper, we essentially settle this problem, showing that the same family is extremal for any k≥ 100 and $n>2k$.