Cross-Sperner families

TL;DR

This paper investigates the maximum sizes of cross-Sperner families, establishing bounds on their product and sum, and characterizing the extremal configurations for large ground set sizes.

Contribution

It provides new bounds on the size of cross-Sperner families and identifies the extremal structures maximizing these measures.

Findings

Product of sizes is at most 2^{2n-4}.

Sum of sizes is maximized when one family contains a single middle-sized set.

Results hold for sufficiently large n with both families non-empty.

Abstract

A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G \subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$: with the sum $|\cF|+|\cG|$ or with the product $|\cF|\cdot |\cG|$. We show that if $\cF, \cG \subseteq 2^{[n]}$, then $|\cF||\cG| \le 2^{2n-4}$ and $|\cF|+|\cG|$ is maximal if $\cF$ or $\cG$ consists of exactly one set of size $\lceil n/2 \rceil$ provided the size of the ground set $n$ is large enough and both $\cF$ and $\cG$ are non-empty.