Show that if each point in the plane is colored with 3 different colors, a unit segment must exist whose endpoints are the same color.
Set up two equilateral triangles ABC and BCD with unit sides which share a side. Let the color of A be green.
If B or C are green then we are done. Likewise if D is the same color as B or C.
This leaves the case where D is green. Since the triangle can be oriented in any direction, all points the same distance from A as D is colored green. This describes a green circle.
The circles has a larger than unit diameter, and thus there are an infinite number of points on the circle which are a unit distance from each other.