A note on short cycles in a hypercube

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 f(n,Cl) be the largest number of edges in a subgraph of a hypercube Qn containing no cycle of length l. It is known that f(n,Cl) = o(|E(Qn)|), when l = 4k, k ≥ 2 and that f(n, C6) ≥ 13 |E(Qn)|. It is an open question to determine f(n,Cl) for l = 4k + 2, k ≥ 2. Here, we give a general upper bound for f(n,Cl) when l = 4k + 2 and provide a coloring of E(Qn) by 4 colors containing no induced monochromatic C10.