Hint

Suppose we wanted to prove something similar but simpler, namely that any colouring must include three equally-spaced points in a line all the same colour. How would we go about it?

First, note that we can certainly find two points the same colour! Pick two, and draw a line through them, marking them as "-1" and "1". This gives a scale on which you can label other points on the line, too. The points we are interested in are -3, 0 and 3.

It's easy to check that however you colour these three points, you will get three equally-spaced points of one colour. You can try it out yourself below.