Elemental Methods in Ergodic Ramsey Theory

by McCutcheon, Randall

• ISBN13: 9783540668091
• ISBN10: 3540668098
• Format: Paperback
• Copyright: 1999-12-01
• Publisher: Springer Verlag

Summary

This book, suitable for graduate students and professional mathematicians alike, didactically introduces methodologies due to Furstenberg and others for attacking problems in chromatic and density Ramsey theory via recurrence in topological dynamics and ergodic theory, respectively. Many standard results are proved, including the classical theorems of van der Waerden, Hindman, and SzemerA(c)di. More importantly, the presentation strives to reflect the extent to which the field has been streamlined since breaking onto the scene around twenty years ago. Potential readers who were previously intrigued by the subject matter but found it daunting may want to give a second look.

Table of Contents

Introduction	p. 1
Ramsey Theory and Topological Dynamics	p. 5
Preliminaries	p. 5
Van der Waerden's theorem	p. 7
Gallai's theorem	p. 11
A polynomial van der Waerden theorem	p. 16
Z[subscript k] and IP van der Waerden theorems	p. 20
The Hales-Jewett coloring theorem	p. 26
Recurrence for VIP-systems	p. 29
The Bergelson-Leibman coloring theorem	p. 33
Infinitary Ramsey Theory	p. 40
Ramsey's theorem and Schur's theorem	p. 40
Hindman's theorem	p. 44
The Carlson-Simpson theorem	p. 50
Carlson's theorem	p. 57
Central sets	p. 62
An infinitary set-polynomial theorem	p. 68
Density Ramsey Theory	p. 78
Measure theoretic preliminaries	p. 78
Furstenberg correspondence	p. 84
Kriz' example	p. 89
Hilbert spaces	p. 93
Sarkozy's theorem	p. 100
Polynomial recurrence along IP-sets	p. 103
Three Ergodic Roth Theorems	p. 111
Ergodicity and weak mixing	p. 111
Roth's theorem	p. 115
Measurable decomposition	p. 119
Two dimensional Roth theorem	p. 123
Double recurrence along IP sets	p. 128
Multiple Recurrence	p. 136
Furstenberg's structure theorem	p. 136
Szemeredi's theorem	p. 141
A polynomial Szemeredi theorem	p. 145
References	p. 153
List of Symbols	p. 157
Index	p. 158
Table of Contents provided by Blackwell. All Rights Reserved.