A painted plane II

Your fears have been realized: I am presenting more painted plane problems!

1.
Let's say I've painted each point on an infinite, continuous plane. Each point is either painted red or blue. Prove that there must exist three points of the same color which form the corners of an equilateral triangle.

2.
I've painted the plane again, this time with three colors. Each point is either red, blue, or green. Prove that there must either exist three points of the same color, or of three different colors, such that the points form the corners of an equilateral triangle.

The solution has now been posted! see here

Oh, and yes, there will be even more painted plane puzzles in the future.