Saturated Subgraphs of the Hypercube

We say a graph is (Q n ,Q m )-saturated if it is a maximal Q m -free subgraph of the n-dimensional hypercube Q n . A graph is said to be (Q n ,Q m )-semi-saturated if it is a subgraph of Q n and adding any edge forms a new copy of Q m . The minimum number of edges a (Q n ,Q m )-saturated graph (respectively (Q n ,Q m )-semi-saturated graph) can have is denoted by sat(Q n ,Q m ) (respectively s-sat(Q n ,Q m )). We prove that \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath} for fixed m, disproving a conjecture of Santolupo that, when m=2, this limit is 1/4. Further, we show by a different method that sat(Q n , Q 2)=O(2 n ), and that s-sat(Q n , Q m )=O(2 n ), for fixed m. We also prove the lower bound \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath} thus determining sat(Q n ,Q 2) to within a constant factor, and discuss some further questions.