the Hilton–Milner family. It is an easy calculation to see that $|\mathcal J_i|>|\mathcal J_{i+1}|$ for every $k\ge 4$ and $i\in [k-1]$.[^1]

\[thmhk\] Let $n>2k$, $k\ge 4$. Then any intersecting family ${\mathcal F}$ with $\gamma({\mathcal F})\ge 2$ satisfies $$\label{eqhk}|{\mathcal F}|\le {n-1\choose k-1}-{n-k-1\choose k-1}-{n-k-2\choose k-2}+2,$$ moreover, for $k\ge 5$ the equality is attained only on the families isomorphic to $\mathcal J_2$.

We note that Han and Kohayakawa also proved their theorem for $k= 3$, as well as described the cases of equality for $k=4$. These cases are more tedious and do not follow from our methods in a straightforward way. However, Theorem \[thmhk\] can be deduced without much effort. A slightly weaker version of the theorem above (without uniqueness) is a consequence of the main result in the paper by Hilton and Milner [@HM] (cf. also [@HK]).

Applying Corollary \[corhm\] with $u=k$, we conclude that holds for $k\ge 4$. The bound is sharp, as witnessed by $\mathcal J_2$. The uniqueness requires hardly more effort, but since it uses Theorem \[thmfull1\], stated in Section \[sec3\], we postpone its proof until Section \[sec4\].

For any set $X$, family ${\mathcal F}\subset 2^X$ and $i\in X$, we use the following standard notations $$\begin{aligned}
{\mathcal F}(\bar i):=&\ 