Two-part set systems

TL;DR

This paper explores variations of the two-part Sperner theorem involving intersection and subset properties, and introduces a new result on intersecting, cross-Sperner families with bounds on their combined size.

Contribution

It proves a novel bound on the sum of sizes of intersecting, cross-Sperner families, and investigates related set family properties in two-part systems.

Findings

Bound on sum of sizes: |    |     < 2^{n-1}

Exponential examples show the tightness of the bound

Generalizations of Sperner and intersection properties in bipartite set systems

Abstract

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\cF$ is a family of subsets of $X$ such that no two sets $A, B \in \cF$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B \cap X_i$ for some $i$, then $|\cF| \le {n \choose \lfloor n/2 \rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\cF, \cG$ be families of subsets of an $n$-element set such that $\cF$ and $\cG$ are both intersecting and cross-Sperner, meaning that if $A \in \cF$ and $B \in \cG$, then $A \not\subset B$ and $B \not\subset A$. Then $|\cF| +|\cG| < 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.