Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initi
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Mauro Di Nasso Isaac Goldbring Martino Lupini
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Editors: Michel Brion, Grenoble Camillo De Lellis, Princeton Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Cambridge Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg
2239
More information about this series at http://www.springer.com/series/304
Mauro Di Nasso • Isaac Goldbring • Martino Lupini
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
123
Mauro Di Nasso Department of Mathematics Universita di Pisa Pisa, Italy
Isaac Goldbring Department of Mathematics University of California, Irvine Irvine, CA, USA
Martino Lupini School of Mathematics and Statistics Victoria University of Wellington Wellington, New Zealand
ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-17955-7 ISBN 978-3-030-17956-4 (eBook) https://doi.org/10.1007/978-3-030-17956-4 Mathematics Subject Classification (2010): Primary: 05D10, Secondary: 03H10 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To Marian, Karina, and Nicola
Preface
Generally speaking, Ramsey theory studies which combinatorial configurations of a structure can always be found in one of the pieces of a given finite partition. More generally, it considers the problem of which combinatorial configurations can be found in sets that are “large” in some suitable sense. Dating back to the foundational results of van der Waerden, Ramsey, Erd˝os, Turán, and ot
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