A painted plane

1.
Let's say I've painted a plane--the flat kind, not the flying kind. Every single point on the plane is assigned the color red or blue. Prove that there must exist two points of the same color that are exactly one unit apart.

2.
Every single point on the plane is assigned the color red, blue or green. Prove that there must exist two points of the same color that are exactly one unit apart.

Update: Yoo Chung has explained the solution here.
My additional comments are here.