  The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere
is covered by d+1 closed sets, then at least one of the sets must contain a
pair of antipodal points. In this paper, we prove a combinatorial version of
this theorem for hypercubes. It is not hard to show that for any cover of the
facets of a d-cube by d sets of facets, at least one such set contains a pair
of antipodal ridges. However, we show that for any cover of the ridges of a
d-cube by d sets of ridges, at least one set must contain a pair of antipodal
k-faces, and we determine the maximum k for which this must occur, for all
dimensions except d=5.
