Painted Plane solution

Congratulations to Yoo Chung of Stochastic Scribbles for solving the Painted Plane puzzle!

He has a very nice and illustrated explanation over here.

If I may add one more illustration, here is an equivalent way to think about it.

Click for spoiler diagram (I stole part of the diagram from Yoo)

We start with the same setup: assume that there are no two points of the same color which are one unit apart. If we start with any red point C, we can deduce that point D is red as well. We can use the same reasoning to show that there must be a red circle with center C and a radius of sqrt(3). There must be two points on this red circle which are exactly one unit apart. Contradiction, ergo our original assumption was incorrect. QED

Sorry if that was a bit dense, but Yoo already wrote a clear enough solution, so there you go.

I've always wondered whether it's possible to do it with 4 colors. But alas, I've never been able to figure it out, and there's no reason to assume the answer is accessible.

This is one of my favorite puzzles of all time. So guess what? I'm going to have some similar puzzles in the future! Mwahahahahaaa!