In this paper we modify slightly Razborov’s flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a -cycle-free subgraph of the -dimensional hypercube is at most times the number of its edges. We also improve the upper bound on the number of edges for -cycle-free subgraphs of the -dimensional hypercube from to times the number of its edges. Additionally, we show that if the -dimensional hypercube is considered as a poset then the maximum vertex density of three middle layers in an induced subgraph without 4-cycles is at most 2.15121.