  In this paper, we address a particular variation of the Tur\'an problem for
the hypercube. Alon, Krech and Szab\'o (2007) asked "In an n-dimensional
hypercube, Qn, and for l < d < n, what is the size of a smallest set, S, of
Q_l's so that every Q_d contains at least one member of S?" Likewise, they
asked a similar Ramsey type question: "What is the largest number of colors
that we can use to color the copies of Q_l in Q_n such that each Q_d contains a
Q_l of each color?" We give upper and lower bounds for each of these questions
and provide constructions of the set S above for some specific cases.
