van der Waerden's Theorem

This chapter is devoted to van der Waerden’s theorem, another of the bedrocks of Ramsey’s theory. The theorem establishes as a fact that any finite colouring of natural numbers contains monochromatic arithmetic progressions of any (finite) length.

The chapter starts with a section about Bartel Leendert van der Waerden’s life and work.

This is followed by a visualized proof of van der Waerden’s theorem. This is done in two steps. The second section of this chapter presents the proof of the theorem in the case of three-term arithmetic progressions. The third section discusses a general case of van der Waerden’s theorem. The concept of van der Waerden numbers and Szemerédi’s theorem are introduced in the next section.

The following section contains reflections about several questions and results that sprout from van der Waerden’s theorem and have marked the development of Ramsey theory.

The end of the chapter includes exercises related to topics presented in this chapter.