GRAPHS AND COMBINATORICS, vol.31, no.1, pp.235-242, 2015 (SCI-Expanded)
In this paper, we address a particular variation of the Turan problem for the hypercube. Alon, Krech and Szab (SIAM J Discrete Math 21:66-72, 2007) asked "In an n-dimensional hypercube, Q (n) , and for a"" < d < n, what is the size of a smallest set, S, of Q (a"") '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 (a"") in Q (n) such that each Q (d) contains a Q (a"") 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.