Suppose every point in a plane is colored either red or blue. Show that there is an equilateral triangle somewhere in the plane whose vertices are all the same color.
Find two points, A and B, that are the same color, say blue. Then make a triangular array of points as shown.
- If C or D are blue, we’re done.
So assume C is red, and D is red. - If E is red, we’re done: △CDE.
So assume E is blue. - If F is blue, we’re done.
So assume F is red. - Now look at G.
If G is blue, we have △AEG.
If G is red, we have △CFG.