case and the case of cross-intersecting families. In particular, it strengthens the results of [@Fra1], [@FT], [@KZ]. One of the main ingredients in the proof of the main result of Section \[sec3\] (as well as in the proofs of Theorems \[thmfr\], \[thm1\]) is the famous Kruskal–Katona theorem [@Kr; @Ka]. This theorem is a central result in extremal set theory and has numerous applications (see, e.g., [@BK; @BT]). Another key ingredient is the bipartite switching trick, which was introduced in [@KZ] (similar ideas appeared earlier in [@FK1]). In this paper, we exploit this trick to a much greater extent.

We do not state the aforementioned theorem in this section since it requires some preparations. Instead, we state the following corollary, which can be seen as a generalization of the Hilton–Milner phenomena.

\[corhm\] Let $n>2k\ge 8$ and ${\mathcal F}\subset {[n]\choose k}$ be an intersecting family. Suppose that $\gamma({\mathcal F})>{n-u-1\choose n-k-1}$ for some integer $4\le u\le k$. Then $$\label{eqhm}|{\mathcal F}|\le {n-1\choose k-1}+{n-u-1\choose n-k-1}-{n-u-1\choose k-1}-{n-k-2\choose k-2}+1.$$ Moreover, the same inequality holds for $n>2k\ge 6$ with $u=3$ if ${n-4\choose k-3}<\gamma({\mathcal F})\le {n-3\choose k-2}-{n-k-2\choose k-2}+1$ or ${n-3\choose k-2}< \gamma({\mathcal F})\le {n-3\choose k-2}+{n-4\choose k-2}-{n-k-2\choose k-2}+1$.

Compare with for integer $u$. The difference in the bounds is ${n-k-2\choose 