Ramsey Theory Yesterday, Today, and Tomorrow
Ramsey theory is a relatively “new,” approximately 100 year-old direction of fascinating mathematical thought that touches on many classic fields of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geomet
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Series Editors H. Bass J. Oesterl´e A. Weinstein
Alexander Soifer Editor
Ramsey Theory Yesterday, Today, and Tomorrow
Editor Alexander Soifer University of Colorado at Colorado Springs College of Letters, Arts, and Sciences 1420 Austin Bluffs Parkway Colorado Springs, CO 80918 USA [email protected]
ISBN 978-0-8176-8091-6 e-ISBN 978-0-8176-8092-3 DOI 10.1007/978-0-8176-8092-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938367 c Springer ScienceCBusiness Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer ScienceCBusiness Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Front cover utilizes the photograph of Frank Plumpton Ramsey (1903–1930), aged 18. Reproduced by kind permission of the Provost and the Scholars of King’s College, Cambridge. www.birkhauser-science.com
Dedicated to Paul Erd˝os, who envisioned Ramsey Theory, shaped much of it by his problems, conjectures and results, and inspired us all to dive into its deep and beautiful waters.
What This Book Is About and How It Came into Being
Ramsey theory is a fascinating, approximately 100-year-old field of mathematics that has a non-empty intersection with combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, measure theory, and so on. Ramsey theory possesses its own unifying ideas, and some of its results are among the most beautiful theorems of mathematics. The main mathematical idea of Ramsey theory is this: no matter how large and elaborate a system S is, and how large a positive integer n is, we can choose a large enough super system Q containing S , so that no matter how Q is colored in n colors, Q contains a monochromatic copy of S . Thus one can say that Ramsey theory studies mathematics of coloring. In 2008 the director, Fred Roberts, and the executive committee of DIMACS1 invited me to organize a three-day workshop on Ramsey theory. In response to Dr. Roberts’ desire to host a nongeneric original view of the field, I proposed Ramsey Theory: Yesterday, Today, and Tomorrow. This was approved, and the workshop took place on May 27–29, 2009 at the Busch Campus of Rutgers University in Piscataway, New Jersey. The workshop looked at the emergence and history of Ramsey theory (Yesterday), its results (Today), and its future (Tomorrow) through its open problems, conjectures, and aspirations. In addition to mathematical and historical research, w
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