## Thursday, May 6, 2010

### Chess Board Coloring

Each cell in an infinite chess board is colored with one of the $n$ available colors. Prove that we can always find a rectangle such that all four corners have the same color.

Advanced version: Suppose not all cells are colored, but only some of them. Furthermore, for any circle with radius R, we can always find a colored cell in that circle. Prove that we can still find a monochromatic rectangle.