complications.

For a family ${\mathcal F}$, let $\tau({\mathcal F})$ denote the of ${\mathcal F}$, that is, the minimum size of set $S$ that intersects all sets in ${\mathcal F}$. Each such $S$ we call a .

Intersecting families of $k$-sets with fixed covering number were studied in several classical works. The Erdős–Ko–Rado theorem shows that the largest intersecting family of $k$-element sets has covering number $1$. The aforementioned result of Hilton and Milner [@HM] determined the largest intersecting family with covering number $2$. It is clear that any $k$-uniform intersecting family ${\mathcal F}$ satisfies $\tau({\mathcal F})\le k$: indeed, any set of ${\mathcal F}$ is a hitting set for ${\mathcal F}$. In a seminal paper [@EL], Erdős and Lovász proved that an intersecting family ${\mathcal F}\subset {[n]\choose k}$ with $\tau({\mathcal F})=k$ has size at most $k^k$ (note that it is independent of $n$!) and provided a lower bound of size roughly $(k/e)^k$. Later, both lower [@FOT2] and upper [@Che; @AR; @Fra21] bounds were improved.

Let us define the following important family. $$\label{deft2}
\mathcal T_2(k):=\big\{[k]\big\}\cup \big\{\{1\}\cup [k+1,2k-1]\big\}\cup \big\{\{2\}\cup [k+1,2k-1]\big\}.$$ It is easy to see that $\mathcal T_2(k)$ is intersecting, moreover, $\tau(\mathcal T_2(k))=2$. In [@F16], Frankl studied the following question: what is the 