On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers.

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph and a hypercube , is the largest number of edges in an -free subgraph of . If is at least a positive proportion of the number of edges in , then is said to have positive Turán density in the hypercube; otherwise it has zero Turán density.