A Density Version of the Hales-Jewett Theorem for K = 3

This chapter discusses the density version of the Hales–Jewett theorem for k = 3 and outlines the main elements of the proof. The method used is “ergodic” and the first step is to show that the theorem in question can be formulated as a statement regarding a certain family of measure preserving transformations acting on a (probability) measure space. The phenomenon that appears in the treatment of Szemeredi's theorem, is that when a family of measure preserving transformations having a certain structure acts on a measure space, a set of positive measure will necessarily return to itself (“recur”) under certain combinations of transformations. This recurrence phenomenon for sets of positive measure is then translated into the appearance of certain patterns in subsets of a sufficiently large structure, provided the density of the subset is bounded.