Lecture Notes in Mathematics 2239 Mauro Di Nasso Isaac Goldbring Martino Lupini Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory Lecture Notes in Mathematics 2239 1 Editors-in-Chief: 2 Jean-Michel Morel, Cachan 3 Bernard Teissier, Paris 4 Advisory Editors: 5 Michel Brion, Grenoble 6 Camillo De Lellis, Princeton 7 Alessio Figalli, Zurich 8 Davar Khoshnevisan, Salt Lake City 9 Ioannis Kontoyiannis, Cambridge 10 Gábor Lugosi, Barcelona 11 Mark Podolskij, Aarhus 12 Sylvia Serfaty, New York 13 Anna Wienhard, Heidelberg 14 UNCORRECTED PROOF More information about this series at http://www.springer.com/series/304 15 UNCORRECTED PROOF Mauro Di Nasso • Isaac Goldbring • Martino Lupini 16 Nonstandard Methods 17 in Ramsey Theory 18 and Combinatorial Number 19 Theory 20 UNCORRECTED PROOF 123 21 Mauro Di Nasso Isaac Goldbring Department of Mathematics Department of Mathematics Universita di Pisa University of California, Irvine Pisa, Italy Irvine, CA, USA 22 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 23 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 24 © Springer Nature Switzerland AG 2019 25 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 26 the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, 27 broadcasting, reproduction on microfilms or in any other physical way, and transmission or information 28 storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology 29 nowUNCORRECTED known or hereafter developed. PROOF30 The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication 31 does not imply, even in the absence of a specific statement, that such names are exempt from the relevant 32 protective laws and regulations and therefore free for general use. 33 The publisher, the authors and the editors are safe to assume that the advice and information in this book 34 are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or 35 the editors give a warranty, express or implied, with respect to the material contained herein or for any 36 errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional 37 claims in published maps and institutional affiliations. 38 This Springer imprint is published by the registered company Springer Nature Switzerland AG. 39 The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface 1 Generally speaking, Ramsey theory studies which combinatorial configurations of 2 a structure can always be found in one of the pieces of a given finite partition. 3 More generally, it considers the problem of which combinatorial configurations 4 can be found in sets that are “large” in some suitable sense. Dating back to the 5 foundational results of van der Waerden, Ramsey, Erdos,˝ Turán, and others from the 6 1920s and 1930s, Ramsey theory has since then had an extraordinary development. 7 On the one hand, many applications of Ramsey theory to numerous other areas 8 of mathematics, ranging from functional analysis, topology, and dynamics to set 9 theory, model theory, and computer science, have been found. On the other hand, 10 results and methods from other areas of mathematics have been successfully applied 11 to establish new results in Ramsey theory. For instance, ergodic theory or the theory 12 of recurrence in measurable dynamics has had a huge impact on Ramsey theory, to 13 the point of giving rise to the research area of “ergodic Ramsey theory.” Perhaps 14 the best known achievement of this approach is the ergodic-theoretic proof of 15 Szemerédi’s theorem due to Furstenberg in the 1980s. In a different (but intimately 16 related) direction, the theory of ultrafilters has been an important source of methods 17 and ideas for Ramsey theory. In particular, the study of topological and algebraic 18 properties of the space of ultrafilters has been used to give short and elegant 19 proofs of deep combinatorial pigeonhole principles. Paradigmatic in this direction 20 is the Galvin–Glazer ultrafilter proof of Hindman’s theorem on sets of finite sums, 21 previously established by Hindman in 1974 via a delicate, purely combinatorial 22 argument. 23 AQ1 Recently,UNCORRECTED a new thread of research has emerged, where PROOF problems in Ramsey 24 theory are studied from the perspective of nonstandard analysis and nonstandard 25 methods. Developed by Abraham Robinson in the 1960s and based on first-order 26 logic and model theory, nonstandard analysis provided a formal and rigorous 27 treatment of calculus and classical analysis via infinitesimals. Such a treatment is 28 more similar in spirit to the approach originally taken in the development of calculus 29 in the seventeenth and eighteenth century and avoids the epsilon-delta arguments 30 that are inherent in its later formalization due to Weierstrass. While this is perhaps its 31 most well-known application, nonstandard analysis is actually much more versatile. 32 v vi Preface The foundations of nonstandard analysis provide us with a method, which we shall 33 call the nonstandard method, that is applicable to virtually any area of mathematics. 34 The nonstandard method has thus far been used in numerous areas of mathematics, 35 including functional analysis, measure theory, ergodic theory, differential equations, 36 and stochastic analysis, just to name a few such areas. 37 In a nutshell, the nonstandard method allows one to extend the given math- 38 ematical universe and thus regard it as contained in a much richer nonstandard 39 universe. Such a nonstandard universe satisfies strong saturation properties which 40 in particular allow one to consider limiting objects which do not exist in the 41 standard universe. This procedure is similar to passing to an ultrapower, and in fact 42 the nonstandard method can also be seen as a way to axiomatize the ultrapower 43 construction in a way that distillates its essential features and benefits, but avoids 44 being bogged down by the irrelevant details of its concrete implementation. This 45 limiting process allows one to reformulate a given problem involving finite (but 46 arbitrarily large) structures or configurations into a problem involving a single 47 structure or configuration which is infinite but for all purposes behaves as though 48 it were finite (in the precise sense that it is hyperfinite in the nonstandard universe). 49 This reformulation can then be tackled directly using finitary methods, ranging from 50 combinatorial counting arguments to recurrence theorems for measurable dynamics, 51 recast in the nonstandard universe. 52 In the setting of Ramsey theory and combinatorics, the application of non- 53 standard methods was pioneered by the work of Keisler, Leth, and Jin from the 54 1980s and 1990s. These applications focused on density problems in combinatorial 55 number theory. The general goal in this area is to establish the existence of 56 combinatorial configurations in sets that are large in the sense that they have 57 positive asymptotic density. For example, the aforementioned celebrated theorem 58 of Szemerédi from 1970 asserts that a set of integers of positive density contains 59 arbitrarily long finite arithmetic progressions. One of the contributions of the 60 nonstandard approach is to translate the notion of asymptotic density on the integers, 61 which does not satisfy all the properties of a measure, into an actual measure in the 62 nonstandard universe. This translation then makes methods from measure theory 63 and ergodic theory, such as the ergodic theorem or other recurrence theorems, 64 available for the study of density problems. In a sense, this can be seen as a version 65 of Furstenberg’s correspondence (between sets of integers and measurable sets in 66 a dynamical system), with the extra feature that the dynamical system obtained 67 perfectly reflects all the combinatorial properties of the set that one started with. 68 TheUNCORRECTED achievements of the nonstandard approach in this area PROOF include the work of 69 Leth on arithmetic progressions in sparse sets, Jin’s theorem on sumsets, as well 70 as Jin’s Freiman-type results on inverse problems for sumsets. More recently, these 71 methods have also been used by Jin, Leth, Mahlburg, and the present authors to 72 tackle a conjecture of Erdos˝ concerning sums of infinite sets (the so-called B + C 73 conjecture), leading to its eventual solution by Moreira, Richter, and Robertson. 74 Nonstandard methods are also tightly connected with ultrafilter methods. This 75 has been made precise and successfully applied in a recent work of one of us (Di 76 Nasso), where he observed that there is a perfect correspondence between ultrafilters 77 Preface vii and elements of the nonstandard universe up to a natural notion of equivalence. On 78 the one hand, this allows one to manipulate ultrafilters as nonstandard points and to 79 use ultrafilter methods to prove the existence of certain combinatorial configurations 80 in the nonstandard universe. On the other hand, this gives an intuitive and direct 81 way to infer, from the existence of certain ultrafilter configurations, the existence of 82 corresponding standard combinatorial configurations via the fundamental principle 83 of transfer from nonstandard analysis. This perspective has successfully been 84 applied by Di Nasso and Luperi Baglini to the study of partition regularity problems 85 for Diophantine equations over the integers, providing in particular a far-reaching 86 generalization of the classical theorem of Rado on partition regularity of systems 87 of linear equations.