A Note On The Cross-Sperner Families

TL;DR

This paper determines the maximum size of the intersection set for cross-Sperner families of subsets of [n], solving an open problem in extremal set theory.

Contribution

It provides an exact formula for the maximum intersection size of cross-Sperner families, resolving a previously open problem.

Findings

Derived the exact maximum intersection size formula for cross-Sperner families.

Proved the conjecture posed by Frankl and Wang.

Solved an open problem in extremal combinatorics.

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 $m(n)={\rm{max}}\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset2^{[n]}~{\rm{are~cross}}$-${\rm{sperner}}\}$, 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.