If each point of the plane is colored red or blue, prove there are two points of the same color at distance one inch from each other.

We shall argue by the pigeon hole principle:

Consider the vertices of an equilateral triangle in the plane whose side length is one inch. There are two colors (our "pigeon-holes") and 3 vertices (our "pigeons"). Two of these vertices must hence be of the same color. Noting that every pair of such vertices are at distance one inch from one another, we are done.