delta-method is one of the reasons.) In particular, the bound on $n$ in Theorem \[thmko\] is doubly exponential in $k$. In this paper, we deduce Theorem \[thmko\] from a much more general result, which, additionally, holds without any restriction on $n$. A family is called with respect to some property, if none of its proper subfamilies possesses the property. For shorthand, we say that ${\mathcal M}$ is if, for any $M_l\in {\mathcal M}$, we have $|\bigcap_{M\in {\mathcal M}\setminus\{M_l\}}M|>|\bigcap_{M\in {\mathcal M}}M|$. The following theorem is one of the main results of this paper.

\[thmclass2\] Assume that $n>2k\ge 8$. Consider an intersecting family ${\mathcal F}\subset{[n]\choose k}$ with $\Delta({\mathcal F}) = d_1({\mathcal F})$. Take a subfamily $\mathcal M\subset {\mathcal F}(\bar 1)$, which is minimal w.r.t. common intersection and such that $|\bigcap_{M\in {\mathcal M}}M|= t$. Take the (unique) maximal intersecting family ${\mathcal F}'$, such that ${\mathcal F}'(\bar 1) = {\mathcal M}$. If $t\ge 3$ then we have $$\label{eqclass1} |{\mathcal F}|\le |{\mathcal F}'|,$$ and, for $k\ge 5$ equality is possible if and only if ${\mathcal F}$ is isomorphic to ${\mathcal F}'$.

Moreover, if ${\mathcal F}$ is as above and $|\bigcap_{F\in {\mathcal F}(\bar 1)}F|\le t$ for some $t\ge 3$, then $$\label{eqclass2} |{\mathcal F}|\le |\mathcal J_{k-t+1}|,$$ and, 