How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdős about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let be the largest number of edges in a subgraph of a hypercube containing no cycle of length l. It is known that , when , and that . It is an open question to determine for , . Here, we give a general upper bound for when and provide a coloring of by four colors containing no induced monochromatic .