Cutsets of Boolean lattices

In graph theory, Menger’s Theorem is the basis of many results concerning notions of connectedness. There are many versions, in this paper the following will be useful. Let G = (V, E) be a directed graph and A, B disjoint subsets of V with no directed edge from A to B. A subset of V -A U B is called an (A, B) separating vertex-set if every directed path from A to B intersects F. A set of directed paths from A to B are called vertex disjoint if any two have at most one vertex of A and one vertex of B in common.