8 S ep 1 99 3 ULTRAFILTERS : WHERE TOPOLOGICAL DYNAMICS = ALGEBRA = COMBINATORICS

We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem. Furstenberg and his co-workers have shown [15, 16, 17] how to deduce combinatorial consequences from theorems about topological dynamics in compact metric spaces. Bergelson and Hindman [4] applied similar methods in non-metrizable spaces, particularly the Stone-Čech compactification βN of the discrete space of natural numbers. This approach and related ideas of Carlson [11] lead to particularly simple formulations since many of the basic concepts of dynamics, when applied to βN, can be expressed in terms of a semigroup operation on βN, the natural extension of addition on N. The semigroup βN can also substitute, in many contexts, for the enveloping semigroups ([14]) traditionally used in topological dynamics. Further simplifications and applications of these ideas were developed in [3]. The purpose of this paper is to survey some of these developments. In contrast to most surveys, however, we include some detailed proofs, in order to emphasize their simplicity. In the first three sections, we develop the necessary theory of dynamics and the equivalent semigroup structure in βN. In the fourth section, we apply the theory to present proofs of Hindman’s partition theorem for finite sums and of the Hales-Jewett theorem about homogeneous combinatorial lines in cubes. A final section (omitted for lack of time in the talk on which this paper is based) compares the ultrafilters discussed in the earlier sections with other ultrafilters traditionally related to combinatorics, for example selective ultrafilters.