Cross-Disjoint Pairs of Clouds in the Interval Lattice

Summary

Let $\mathcal{I}_{n}$ be the lattice of intervals in the Boolean lattice $\mathcal{L}_{n}$. For $\mathcal{A},\mathcal{B}\,\subset \,\mathcal{I}_{n}$ the pair of clouds $(\mathcal{A},\mathcal{B})$ is cross-disjoint, if $I \cap J = \emptyset$ for $I \in \mathcal{A},\ J \in \mathcal{B}$. We prove that for such pairs $\vert \mathcal{A}\vert \vert \mathcal{B}\vert \leq {3}^{2n-2}$ and that this bound is best possible.

Optimal pairs are up to obvious isomorphisms unique. The proof is based on a new bound on cross intersecting families in $\mathcal{L}_{n}$ with a weight distribution. It implies also an Intersection Theorem for multisets of Erdős P, Schőnheim J (1969) On the set of non pairwise coprime division of a number. In: Proc. of the Colloquium on Comb. Math. Dalaton Füred, pp 369–376.