The Theorems of van der Waerden and Hales-Jewett

Abstract

As we have seen in the previous chapter, Ramsey’s theorem proved to be crucial in the development of Ramsey theory, to which it gave its name. However, Ramsey theory has another equally important root in works of Hindman, Shur, and van der Waerden motivated by problems about rational functions and modular arithmetic: Hilbert’s Cuble Lemma, Schur’s Lemma, and van der Waerden’s Theorem on arithmetic progressions. Particularly, the latter is of fundamental importance, as it paved the way to many of the later developments in Ramsey theory, including partition regularity of diophantine equations (see Chap. 9) and density results in additive combinatorics (see Chap. 10). The combinatorial essence of van der Waerden’ theorem was later isolated by Hales and Jewett, who proved a powerful abstract pigeonhole principle, later generalized further by Graham and Rothschild. In this chapter, we will present nonstandard proofs of both van der Waerden’s Theorem and the Hales–Jewett theorem.