MATCHINGS AND PATHS IN THE CUBE

In this note we are concerned with the existence of matchings and fami lies of disjoint paths between subsets of the 6-dimensional discrete c ube Q(n). For example, we show that if A is a subset od Q(n) of size S igma(i=0)(k)((n)(i)), where k < 1/2n, then there is a matching from A to its complement of size at least ((n)(k)). We also present a conject ure concerning the existence of directed paths, and prove some related results.