terms. For a family ${\mathcal F}$, the $\gamma({\mathcal F})$ is the quantity $|{\mathcal F}|-\Delta({\mathcal F})$. One may think of diversity as of the distance from ${\mathcal F}$ to the closest star.

The following strengthening of Theorem \[thmfr\] was obtained by Kupavskii and Zakharov [@KZ].

\[thm1\] Let $n>2k>0$ and ${\mathcal F}\subset {[n]\choose k}$ be an intersecting family. If $\gamma({\mathcal F})\ge {n-u-1\choose n-k-1}$ for some real $3\le u\le k$, then $$\label{eq01}|{\mathcal F}|\le {n-1\choose k-1}+{n-u-1\choose n-k-1}-{n-u-1\choose k-1}.$$

We note that the Hilton–Milner theorem, as well as Theorem \[thmfr\], is immediately implied by Theorem \[thm1\]. The deduction of Theorem \[thm1\] from Theorem \[thmfr\] for integer values of $u$ is possible, but not straightforward. Theorem \[thm1\] provides the strongest known stability result for the Erdős–Ko–Rado theorem for large intersecting families, more precisely, for the families of size at least ${n-2\choose k-2}+2{n-3\choose k-2}$. There are several other stability results for the Erdős–Ko–Rado theorem, see, e.g. [@DT; @EKL; @Fri].

In Section \[sec3\], we prove a conclusive version of Theorem \[thm1\], which gives the precise dependence of size of an intersecting family ${\mathcal F}$ of $k$-sets on (the lower bound on) $\gamma({\mathcal F})$. The result then is extended to cover the equality case, as well as the weighted 