bound in general for larger $\gamma$. Indeed, the intersecting family $\mathcal H_2$ (cf. ) attains the bound above on the cardinality and has diversity ${n-3\choose k-2}$. The second part of Corollary \[corhm\] complements Theorem \[thm1\] in this respect, showing the aforementioned example is essentially the only exception. We also note that in a recent work [@Kup21] the author managed to prove that there exists $c$, such that for any $n,k$ satisfying $n\ge ck$ we have $\gamma({\mathcal F})\le {n-3\choose k-2}$ for any intersecting ${\mathcal F}\subset{[n]\choose k}$. Earlier, this was shown to be true for $n\ge 6k^2$ by Frankl [@Fra6]. Thus, for $n\ge ck$ Theorem \[thmfull1\] gives a complete answer to the question we address.

In Section \[sec33\], we shall analyze the cases when the equality in can be attained, as well as the cases of very large diversity. In particular, we shall prove the $u=3$ part of Corollary \[corhm\].

Some examples and Theorem \[thmfull1\] restated {#sec31}
-----------------------------------------------

Let us try to familiarize the reader with the statement of Theorem \[thmfull1\]. We have $\gamma_i = i$ for $i = 1,\ldots, k-3$. Indeed, for $1\le \gamma <n-k-1$ we have $\gamma  = {n-k-1\choose n-k-1}+{n-k-2\choose n-k-2}+\ldots +{n-k-\gamma\choose n-k-\gamma}$. Thus, for any such $\gamma$ we have 