For what it's worth, the Green-Tao theorem uses similar techniques and is a kind of harder version of a theorem which says that for any n, if you color the inte

For what it's worth, the Green-Tao theorem uses similar techniques and is a kind of harder version of a theorem which says that for any n, if you color the integers in n colors, there will always be some color for which there are arbitrarily long arithmetic progressions.