Induced subgraphs of hypercubes

Let _Qk$Q_k$ denote the k$k$-dimensional hypercube on 2^k$2^k$ vertices. A vertex in a subgraph of _Qk$Q_k$ is full   if its degree is k$k$. We apply the Kruskal–Katona Theorem to compute the maximum number of full vertices an induced subgraph on n≤2^k$n\le 2^k$ vertices of _Qk$Q_k$ can have, as a function of k$k$ and n$n$. This is then used to determine min(max(|V(_H1)|,|V(_H2)|))$min(max(\left|V\left(H_1\right)\right|,\left|V\left(H_2\right)\right|))$ where (i) _H1$H_1$ and _H2$H_2$ are induced subgraphs of _Qk$Q_k$, and (ii) together they cover all the edges of _Qk$Q_k$, that is E(_H1)∪E(_H2)=E(_Qk)$E\left(H_1\right)\cup E\left(H_2\right)=E\left(Q_k\right)$.