Matchings and paths in the cube

AbstractIn this note we are concerned with the existence of matchings and families of disjoint paths between subsets of the n-dimensional discrete cube Qn. For example, we show that if A is a subset of Qn of size ∑ki = 0(ni), where k < 12n, then there is a matching from A to its complement of size at least (ni).We also present a conjecture concerning the existence of directed paths, and prove some related results