For a set $X$, (lex) $\prec$ on the sets from ${X\choose k}$ is a total order, in which $A\prec B$ iff the minimal element of $A\setminus B$ is smaller than the minimal element of $B\setminus A$. For $0\le m\le {|X|\choose k}$ let $\mathcal L(X,m,k)$ be the collection of $m$ largest $k$-element subsets of $X$ with respect to lex. We say that two families $\mathcal A,\mathcal B$ are if $A\cap B\ne \emptyset$ for any $A\in\mathcal A, B\in \mathcal B$.

\[thmHil\]If $\mathcal A\subset {[n]\choose a}, \mathcal B\subset {[n]\choose b}$ are cross-intersecting then the families $\mathcal L([n],|\mathcal A|,a),\mathcal L([n],|\mathcal B|, b)$ are also cross-intersecting.

In this section, we analyze in great details the relationship between the diversity of an intersecting family and its size. We first note that, if the value of diversity is given precisely, then it is easy to determine the largest intersecting family with such diversity. Indeed, the subfamilies of sets containing the element of the largest degree and not containing the element of the largest degree are cross-intersecting, and one can get exact bounds using Theorem \[thmHil\]. Studying the size of an intersecting family with given upper bounds on diversity is not interesting: the largest intersecting family has diversity 