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We prove some properties of the Kruskal-Katona function, and apply to thefollowing variation of cross-intersecting antichains. Let $n\ge 4$ be an eveninteger and $\mathscr{A}$ and $\mathscr{B}$ be two cross-intersectingantichains of $\mathbb{N}_n$ with at most $k$ disjoint pairs, i.e. for all$A_i\in \mathscr{A}$, $B_j\in\mathscr{B}$, $A_i\cap B_j=\emptyset$ only if$i=j\le k$. We prove a best possible upper bound on$|\mathscr{A}|+|\mathscr{B}|$. Furthermore, we show that the extremal familiescontain only $\frac{n}{2}$ and $(\frac{n}{2}+1)$-sets.

Kruskal-Katona's function and a variation of cross-intersecting antichains