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)f(n,Cl) be the largest number of edges in a subgraph of a hypercube QnQn containing no cycle of length l  . It is known that f(n,Cl)=o(|E(Qn)|)f(n,Cl)=o(|E(Qn)|), when l=4kl=4k, k⩾2k⩾2 and that f(n,C6)⩾13|E(Qn)|. It is an open question to determine f(n,Cl)f(n,Cl) for l=4k+2l=4k+2, k⩾2k⩾2. Here, we give a general upper bound for f(n,Cl)f(n,Cl) when l=4k+2l=4k+2 and provide a coloring of E(Qn)E(Qn) by four colors containing no induced monochromatic C10C10.