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In this section we obtain the concluding version of Theorem \[thm1\], which tells [*exactly*]{}, how large an intersecting family may be, given a [*lower*]{} bound on its diversity. We determine all “extremal” values of diversity and the sizes of the corresponding families.

The difficulty to obtain such a version of Theorem \[thm1\] is that, while Theorem \[thmHil\] gives a very strong and clear characterisation of families with fixed diversity, the size of the family is not monotone w.r.t. diversity (the size of the largest family with given diversity does not necessarily decrease as diversity increases, although it is true in “most” cases). Moreover, the numerical dependence between maximal possible sizes of ${{\mathcal A}}$ and ${{\mathcal B}}$ given by Theorem \[thmHil\] is complicated and difficult to work with, see [@FMRT] and [@FT7]. Thus, an effort is needed to find the right point of view on the problem.+0.1cm

We shall give two versions of the main theorem of this section. First, we give a “quantitative” version with explicit sharp bounds on the size of intersecting families depending on the lower bound on their diversity. It may be more practical to apply in some cases, but it is difficult to grasp what 