Suppose you have an infinite plane, and each point on the plane has been arbitrarily painted one of two colors.

Prove that there exists an equilateral triangle whose vertices are all the same color.

What is the fewest number of points needed to prove this?

(In reply to 5-point proof by Bryan)

Brian
I believe that the problem is asking for the fewest number of points so that no matter how you rearrange the colors you will always have an equilateral triangle with all of the same color points. If you place the two different colors in different positions but still keeping the five points you have there isn't an equilateral triangle meeting the condition. For example you can rearrange them like this:
r
b r b
r
There is no equilateral triangle meeting the conditions in this arrangement.

Posted by Kelsey on 2003-08-26 15:29:18