On van der Waerden's theorem on arithmetic progressions (Q1160181)

The author presents a proof of the van der Waerden Theorem on arithmetic progressions using divisions into two classes only, but expanding arithmetic progressions to higher dimensions. An essential of other proofs is Artin's idea to generalize the claim to divisions into any finite number of classes. [Reviewer's Remark: The author and the reviewer are in the consensus that in the definition of a canonical subprogression \(Y\) of an arithmetic progression \(X\) each element of \(X\) must be allowed to be the first element of \(Y\).]