k-2}-1$, while the lower bounds on diversity differ by $1$. Thus, Corollary \[corhm\] states that the size of the largest family with diversity at least $\gamma$ has a big drop when $\gamma$ passes the point ${n-u-1\choose n-k-1}$ for integer $u$. We also note that Corollary \[corhm\] is sharp, as it will be clear from Section \[sec3\].\
Numerous authors aimed to determine [*precisely*]{}, what are the largest intersecting families in ${[n]\choose k}$ with certain restrictions. One of such questions was studied by Han and Kohayakawa, who determined the largest family of intersecting families that is neither contained in the Erdős–Ko–Rado family, nor in the Hilton–Milner family. In our terms, the question can be simply restated as follows: what is the largest intersecting family with $\gamma({\mathcal F})\ge 2$? The proof of Han and Kohayakawa is quite technical and long. Kruskal–Katona-based arguments allow for a very short and simple proof in the case $k\ge 5$. For $i\in[k]$ let us put $I_i:=[i+1,k+i]$ and $$\mathcal J_i:=\ \{I_1,I_i\}\cup \Big\{F\in{[n]\choose k}\ :\ 1\in F, F\cap I_1\ne \emptyset, F\cap I_i\ne \emptyset\Big\}.$$ We note that $\mathcal J_i\subset{[n]\choose k}$ and that $\mathcal J_i$ is intersecting for every $i\in [k]$. Moreover, $\gamma(\mathcal J_i) = 2$ for $i>1$ and $\mathcal J_1$ is 