ABSTRACT
Hales-Jewett theorem is another of the landmarks in the development of Ramsey theory. The the- orem was inspired by Rado’s notion of regularity, van der Waerden’s theorem, and a generalization of tic-tac-toe, a well-known children’s game.
The first section of this chapter introduces the notion of combinatorial lines. The section justifies the “line” part in the name of these intriguing objects, but also provides evidence that a combinatorial line is something very different from a Euclidean line.
The second section describes a generalization of the tic-tac-toe, and hence provides a motivation for the study of combinatorial lines. The following section contains a visualized proof of the Hales-Jewett theorem. As an example, it is shown that van der Wearden’s theorem is a corollary of the Hales-Jewett theorem.
The last section includes exercises related to the notion of combinatorial lines and some applications of Hales-Jewett theorem.
