Disjoint Cross Intersecting Families

For positive integers $n$ and $r$ such that $r\le \lfloor n\mathrm{/}2\rfloor$, let $X$ be a set of $n$ elements and let $\left(\genfrac{}{}{0px}{}{X}{r}\right)$ be the family of all $r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of $\left(\genfrac{}{}{0px}{}{X}{r}\right)$ are called cross intersecting if $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. One of main tools in the study of extremal set theory, and cross intersecting families in particular, is compression operation. In this paper, we give an example of cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ that the compression operation is not applicable when $\mathcal{A}$ and $\mathcal{B}$ are disjoint. We develop new technique to prove that, for disjoint cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ of $\left(\genfrac{}{}{0px}{}{X}{r}\right)$, $|$\mathcal{A}$| + |$\mathcal{B}$| \leq \binom{n}{r} - \binom{l}{p}$where$n = 2r + l$and$p = min{r, \lceil \frac{l}{2}\rceil}$. This bound is asymptotically sharp.