On a Covering Problem in the Hypercube

In this paper, we address a particular variation of the Turán problem for the hypercube. Alon, Krech and Szabó (2007) asked “In an n-dimensional hypercube, Qn, and for ` < d < n, what is the size of a smallest set, S, of Q`’s so that every Qd 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` in Qn such that each Qd contains a Q` 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.