  How many edges can a quadrilateral-free subgraph of a hypercube have? This
question was raised by Paul Erd\H{o}s about $27$ years ago. His conjecture that
such a subgraph asymptotically has at most half the edges of a hypercube is
still unresolved. Let $f(n,C_l)$ be the largest number of edges in a subgraph
of a hypercube $Q_n$ containing no cycle of length $l$. It is known that $f(n,
C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\geq 2$ and that $f(n, C_6) \geq
\frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for
$l=4k+2$, $k\geq 2$. Here, we give a general upper bound for $f(n,C_l)$ when
$l=4k+2$ and provide a coloring of $E(Q_n)$ by $4$ colors containing no induced
monochromatic $C_{10}$.
