1 - Ergodic Ramsey Theory  pp. 1-62

Ergodic Ramsey Theory

By V. Bergelson

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This survey is an expanded version and elaboration of the material presented by the author at the Workshop on Algebraic and Number Theoretic Aspects of Ergodic Theory which was held in April 1994 as part of the 1993/1994 Warwick Symposium on Dynamics ofZn-actions and their connections with Commutative Algebra, Number Theory and Statistical Mechanics. The leitmotif of this paper is: Ramsey theory and ergodic theory of multiple recurrence are two beautiful, tightly intertwined and mutually perpetuating disciplines. The scope of the survey is mostly limited to Ramsey-theoretical and ergodic questions about Zn-partly because of the proclaimed goals of the Warwick Symposium and partly because of the author's hope that Zn-related combinatorics, number theory and ergodic theory can serve as an ideal lure through which the author's missionary zeal will reach as wide an audience of potential adherents to the subject as possible.

To compensate for the selective neglect of details and for the lack of full generality in some of the proofs, which were imposed by natural time and space limitations, a significant effort was spent on accentuation and motivation of ideas which lead to conjectures and techniques on which the proofs of conjectures hinge.

Here now is a brief description of the content of the five sections constituting the body of this survey. In Section 1 three main principles of Ramsey theory are introduced and their connection with the ergodic theory of multiple recurrence is emphasized.