Two-part 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