A painted plane III

The painted plane returns! See previous parts: part I and part II.

1.
I've painted a flat, infinite, continuous plane. Each point on the plane is either painted red or blue. Prove that there must exist four points of the same color which form the corners of a rectangle.

2.
I've painted the plane again, this time using a large (but not infinite) number of colors. Prove that there must exist four points of the same color which form the corners of a rectangle.

Update: The solution has been posted already.