ground set, it must have the form $\mathcal H_{k}$, where for integer $2\le u\le k$ $$\label{eqhu}\mathcal H_u:=\ \Big\{A\in {[n]\choose k}\ :\ [2,u+1]\subset A\Big\}\cup\Big\{A\in{[n]\choose k}\ :\  1\in A, [2,u+1]\cap A\ne \emptyset\Big\}.$$ $\mathcal J_1$ has size ${n-1\choose k-1}-{n-k-1\choose k-1}+1$, which is much smaller than ${n-1\choose k-1}$, provided $n$ is large as compared to $k$.

For a family ${\mathcal F}\subset 2^{[n]}$ and $i\in [n]$, the $d_i({\mathcal F})$ of $i$ in ${\mathcal F}$ is the number of sets from ${\mathcal F}$ containing $i$. Let $\Delta({\mathcal F})$ stand for the of an element in ${\mathcal F}$. Frankl [@Fra1] proved the following far-reaching generalization of the Hilton–Milner theorem.

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

One can deduce the Hilton–Milner theorem from the $u=k$ case of Theorem \[thmfr\]. Theorem \[thmfr\] is sharp for integer values of $u$, as witnessed by the example . On a high level, it provides us with an upper bound on $|{\mathcal F}|$ in terms of the size of the largest trivial subfamily ([*star*]{}) in ${\mathcal F}$. Let us state a stronger version of Theorem \[thmfr\] in dual 