Hales-Jewett and a generalized van der Warden Theorems

I Can't Believe It's Not Random!

The Hales-Jewett theorem is one of the most fundamental results in Ramsey theory, implying the celebrated van der Waerden theorem on arithmetic progressions, as well an its multidimensional and IP versions. One interesting property of the Hales-Jewett’s theorem is that it is a set theoretical statement, having no structure, and hence making it versatile for applications. Recently I realized that there exists an equivalent formulation of this theorem using some algebraic structure, and indeed it can be seen as an analogue of van der Waerden’s theorem. The main purpose of this post is to establish that equivalence. In an initial section I present the deduction of the multidimensional van der Waerden from the Hales-Jewett theorem, both to set the mood and to set the stage to later establish the analogy between van der Waerden’s theorem and the equivalent formulation of the Hales-Jewett theorem.

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