A Note On The Cross-Sperner Families

Abstract

Let $(\mathcal{F},\mathcal{G})$ be a pair of families of $[n]$, where $[n]=\{1,2,...,n\}$. If $A\not\subset B$ and $B\not\subset A$ hold for all $A\in \mathcal{F}$ and $B\in \mathcal{G}$, then $(\mathcal{F},\mathcal{G})$ is called a Cross-Sperner pair. P. Frankl and Jian Wang introduced the extremal problem that $m(n)=\max \{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset 2^{[n]}\mathrm{are}\mathrm{cross}$-$\mathrm{sperner}\}$, where $\mathcal{I}(\mathcal{F},\mathcal{G})=\{A\cap B:A\in \mathcal{F},B\in \mathcal{G}\}$. In this note, we prove that $m(n)=2^n-2^{\lfloor \frac{n}{2}\rfloor }-2^{\lceil \frac{n}{2}\rceil }+1$ for all $n>1$. This solves an open problem proposed by P. Frankl and Jian Wang.

We solve an open problem proposed by P. Frankl and Jian Wang