Suppose we have an infinite plane, each point colored in either red or blue. Can one find a simple closed curve whose points have the same color?
If so, can we generalize this result into $n$-colored plane? If not, what coloring would give us a counterexample?
Colour point $(x,y)$ red if $x$ is rational, blue otherwise. A monochromatic curve would have to be contained in a vertical line, and couldn't be a simple closed curve.