Point Coloring II

Assume to the contrary that no two points separated by a unit distance have the same colour.

Draw a unit equilateral triangle $ABC$ somewhere in the plane. Its three vertices must have different colours, so they're red, green and blue in some order. Reflect the point $A$ in the line $BC$ to form the new point $A'$ . Then $A'BC$ is also a unit equilateral triangle whose points are different colours; so $A$ and $A'$ - separated by a distance $\sqrt3$ - must be the same colour.

This is true for all points separated by a distance of $\sqrt3$ (we can always construct these two equilateral triangles).

So now we just construct a triangle with sides $\sqrt3$ , $\sqrt3$ and $1$ . From the above, all three points have the same colour; but two of them are separated by a unit distance; contradiction! So we can always find such points.