Elemental Methods in Ergodic Ramsey Theory
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 e
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1722
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1722
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Randall McCutcheon
Elemental Methods in Ergodic Ramsey Theory
Springer
Author Randall McCutcheon Department of Mathematics University of Maryland College Park MD 20742, USA E-mail:[email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
MacCutcheon, Randall: Elemental methods in ergodic Ramsey theory / Randall McCutcheon. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Lecture notes in mathematics; 1722) ISBN 3-540-66809-8
Mathematics Subject Classification (1991): Primary: 28D05; Secondary: 54H20,05DIO, 1IB05, 05A18 ISSN 0075-8434 ISBN 3-540-66809-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 1070335 41/3143-543210
Contents
Introduction 1 Ramsey Theory and Topological Dynamics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Preliminaries Van der Waerden's theorem Gallai's theorem A polynomial van der Waerden theorem Zk and IP van der Waerden theorems The Hales-Jewett coloring theorem Recurrence for VIP-systems The Bergelson-Leibman coloring theorem
2 Infinitary Ramsey Theory 2.1 2.2 2.3 2.4 2.5 2.6
Ramsey's theorem and Schur's theorem Hindman's theorem The Carlson-Simpson theorem Carlson's theorem Central sets An infinitary set-polynomial theorem
3 Density Ramsey Theory 3.1 3.2 3.3 3.4 3.5 3.6
Measure theoretic preliminaries Furstenberg correspondence Kriz' example Hilbert spaces Sarkozy's theorem Polynomial recurrence along IP-sets
1 5 5 7 11 16 20 26 29 33 40 40 44 50 57 62 68 78 78 84 89 93 100 103
4 Three Ergodic Roth Theorems 4.1 4.2 4.3 4.4 4.5
111 111 115 119 123 128
Ergodicity and weak mixing Roth's theorem Measurable decomposition Two dimensional Roth theorem Double recurrence along IP sets
5 Multiple Recurrence 5.1 Furstenberg's structure theorem 5.2 Szemeredi's theorem 5.3 A polynomial Szerneredi theorem
,
136 136 141 145
References
153
List of Symbols
157
Index
158
Introduction These notes, which are based on a graduate course and seminars given at Wesleyan University in the fall of 1997, contain many of the main results of chromatic and density R
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