then $$\label{eqfull1}|{\mathcal F}|\le {n-a_1\choose n-k}+{n-a_{2}\choose n-k-1}+\ldots +{n-a_{s_a}\choose n-k-s_a+1}+\gamma_l,$$ where $\{b_1,\ldots, b_{s_b}\} = T_{\gamma_l}$ and $\{a_1,\ldots, a_{s_a}\} = S_{\gamma_l}$.

The expression in the right hand side of strictly decreases as $l$ increases.

Moreover, the presented bound is sharp: for each $l =1,\ldots, m$ there exists an intersecting family with diversity $\gamma_l$ which achieves the bound in .

Theorem \[thm1\] and the concept of diversity was successfully used to advance in several problems concerning intersecting families [@Feg; @FK5; @IK; @Kup22]. However, sometimes the statement of Theorem \[thm1\] was not fine-grained enough for the applications, and some extra work was needed to deduce some rudimentary versions of Theorem \[thmfull1\]. (One example of an application of an easy application of Theorem \[thmfull1\] is Theorem \[thmhk\] in the introduction.) This is one of our motivations for proving Theorem \[thmfull1\]. The other motivation is the desire to have the conclusive version of Theorems \[thmfr\] and \[thm1\] and highlight the parallel with the original version of the Kruskal–Katona theorem.

Let us mention that we state Theorem \[thmfull1\] only for $\gamma({\mathcal F})\le {n-4\choose k-3}$, since Theorem \[thm1\] already gives us the bound $|{\mathcal F}|\le {n-2\choose k-2}+2{n-3\choose k-2}$ if $\gamma({\mathcal F})\ge {n-4\choose k-3}$, and we cannot get any better 