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 re(2): 5-point proof - to DJ by aln)

The proof involves not randomly choosing two points, but choosing whatever two points are necessary to get the sought triangle. If in fact all the points are the same color (it's really a 1-color plane) then every triangle has vertices all the same color. But if the plane has two colors, it is possible to select two points of different colors. The key is select, not randomly.

Posted by Charlie on 2003-09-21 16:34:10