A canonical restricted version of van der Waerden's theorem (Q1093636)

The following version of van der Waerden Theorem for sequences of integers not containing an arithmetic progression is proved. Theorem. For every integer k there is a set S of integers containing no \((k+1)\)-term arithmetic progression, and such that for any partition of S into any number of classes, there will be either a k-term arithmetic progression in a single class, or a k-term arithmetic progression with each term in a distinct class. It is pointed out that the method of proving of the theorem mentioned above can be applied to prove some other versions of van der Waerden Theorem.