An old (rather easy) contest problem reads as follows:

Each point in a plane is painted one of two colors. Prove that there exist two points exactly one unit apart that are the same color.

This proof can be easily written by constructing an equilateral triangle of side length $1$ unit and asserting that it is impossible for the colors of all three vertices to be pairwise unequal.

However, I was curious about the trickier problem

Each point in a plane is painted one of

threecolors. Do there exist two points exactly one unit apart that are the same color?

...now, if this happened in $3$-space, I could construct a tetrahedron... but I can't do this in $2$-space. Does this not work with three colors, or is the proof just more complicated? If it doesn't work, how can I construct a counterexample?