  We prove several results regarding edge-colored complete graphs and rainbow
cycles, cycles with no color appearing on more than one edge. We settle a
question posed by Ball, Pultr, and Vojt\v{e}chovsk\'{y} by showing that if such
a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd,
then it also does not contain a rainbow cycle of length $m$ for all $m$ greater
than $2n^2$. In addition, we present two examples which demonstrate that this
result does not hold for even $n$. Finally, we state several open problems in
the area.
