for $k\ge 5$, equality is possible only if ${\mathcal F}$ is isomorphic to $\mathcal J_{k-t+1}$.

This theorem generalizes Theorems \[thmhk\] and \[thmko\] and gives a reasonable classification of [*all*]{} large intersecting families. We also note that we cannot in general replace the condition on $t$ by $t\ge 2$. Indeed, one can see that the family $\mathcal H_2$ (cf. ) is much larger than $\mathcal J_{k-1}$ for large $n$.[^4]

\[cormk\] The statement of Theorem \[thmko\] is valid for any $n>2k\ge 10.$

Fix any ${\mathcal F}$ as in Theorem \[cormk\] and assume that $\Delta({\mathcal F}) = d_1({\mathcal F})$. First assume that $|\bigcap_{F\in{\mathcal F}(\bar 1)}|\le k-2$. Since $k\ge 5$, we are in position to apply the second part of Theorem \[thmclass2\] to ${\mathcal F}$, and get a contradiction with $|{\mathcal F}|>|\mathcal J_3|$. Therefore, $|\bigcap_{F\in {\mathcal F}(\bar 1)}F|=k-1$ and thus ${\mathcal F}(\bar 1)$ is isomorphic to $\mathcal E_l(\bar 1)$ for $l = |\mathcal F(\bar 1)|$, which concludes the proof.

The proof of Theorem \[thmclass2\] is given in Section \[sec41\]. The main tool of the proof of Theorem \[thmclass2\] is again the aforementioned bipartite switching trick. In this paper, we exploit it to much greater extent than in [@FK1] and [@KZ]. One key observation that 