A Note on the Subcubes of the n-Cube

In the year 1976, Sergiu Hart considered and solved the following combinatorial problem: given two parameters k and n, find a set of k vertices in the n-cube which has a maximal number of interconnecting edges. Here we consider the more general problem with q-dimensional induced subcubes (for some fixed but arbitrary choice of q) instead of the interconnecting edges. In a different, but provably equivalent formulation, this problem has been solved quite recently but the proof that had been given is complicated. In this paper, we present a direct and comparably simple proof for the extension of Hart's result.