Mauro Di Nasso, Isaac Goldbring, and Martino Lupini Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory August 7, 2017 Springer To... Acknowledgements To be filled in later... vi Introduction Generally speaking, Ramsey theory studies which combinatorial configurations of a structure are 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, Erdos,˝ Turan´ and others from the 1920s and 1930s, Ramsey theory has since then had an extraordinary development. On the one had, many applications of Ramsey theory have been found to numerous other areas of mathematics, ranging from functional analysis, topology, and dynamics, to set theory, model theory, and computer science. On the other hand, results and methods from other areas of mathematics have been successfully applied to establish new results in Ramsey theory. For instance, ergodic theory and the theory of recurrence in measurable dynamics has had a huge impact on Ramsey theory, to the point of giving rise to the research area of “ergodic Ramsey theory.” Perhaps the best known achievement of this approach is the ergodic-theoretic proof of Szemeredi’s´ theorem due to Furstenberg in the 1980s. In a different (but intimately related) direction, the theory of ultrafilters has been an important source of methods and ideas for Ramsey theory. In particular, the study of topological and algebraic properties of the space of ultrafilters has been used to give short and elegant proofs of deep combinatorial pigeonhole principles. Paradigmatic in this direction is the Galvin–Glazer ultrafilter proof of Hindman’s theorem on sets of finite sums, previously established by Hindman in 1974 via a delicate, purely combinatorial argument. Recently, a new thread of research has emerged, where problems in Ramsey theory are studied from the perspective of nonstandard analysis and nonstandard methods. Developed by Abraham Robinson in the 1960s and based on first order logic and model theory, nonstandard analysis provided a formal and rigorous treatment of calculus and classical analysis via infinitesimals, an approach more similar in spirit to the approach originally taken in the development of calculus in the 17th and 18th century, and avoids the epsilon-delta arguments that are inherent in its later formalization due to Weierstrass. While this is perhaps its most well known application, nonstandard analysis is actually much more versatile. The foundations of nonstandard analysis provide an approach, which we shall call the nonstandard method, that is applicable to virtually any area of mathematics. The nonstandard method has thus far been used in numerous areas of mathematics, including functional analysis, measure theory, ergodic theory, differential equations, and stochastic analysis, just to name a few such areas. In a nutshell, the nonstandard method allows one to extend the given mathematical universe and thus regard it as contained in a much richer nonstandard universe. Such a nonstandard universe satisfies strong saturation properties which in particular allow one to consider limiting objects which do not exist in the standard universe. This procedure is similar to passing to an ultrapower, and in fact the nonstandard method can also be seen as a way to axiomatize the ultrapower construction in a way that distillates its essential features and benefits, but avoids being bogged down by the irrelevant details of its con- crete implementation. This limiting process allows one to reformulate a given problem involving finite (but arbitrarily large) structures or configurations into a problem involving a single structure or configuration which is infinite but for all purposes behaves as though it were finite (in the precise sense that it is hyperfinite in the nonstandard universe). This reformulation can then be tackled directly using finitary methods, ranging from combinatorial counting arguments to recurrence theorems for measurable dynamics, recast in the nonstandard universe. In the setting of Ramsey theory and combinatorics, the application of nonstandard methods had been pioneered by the work of Keisler, Leth, and Jin from the 1980s and 1990s. These applications had focused on density problems in combinatorial number theory. The general goal in this area is to establish the existence of combinatorial configurations in sets that are large in that sense that they have positive asymptotic density. For example, the aforementioned celebrated theorem of Szmeredi´ from 1970 asserts that a set of integers of positive density contains arbitrarily long finite arithmetic progressions. One of vii viii Introduction the contributions of the nonstandard approach is to translate the notion of asymptotic density on the integers, which does not satisfies all the properties of a measure, into an actual measure in the nonstandard universe. This translation then makes methods from measure theory and ergodic theory, such as the ergodic theorem or other recurrence theorems, available for the study of density problems. In a sense, this can be seen as a version of Furstenberg’s correspondence (between sets of integers and measurable sets in a dynamical system), with the extra feature that the dynamical system obtained perfectly reflects all the combinatorial properties of the set that one started with. The achievements of the nonstandard approach in this area include the work of Leth on arithmetic progressions in sparse sets, Jin’s theorem on sumsets, as well as Jin’s Freiman-type results on inverse problems for sumsets. More recently, these methods have also been used by Jin, Leth, Mahlburg, and the present authors to tackle a conjecture of Erdos˝ concerning sums of infinite sets (the so-called B +C conjecture). Nonstandard methods are also tightly connected with ultrafilter methods. This has been made precise and successfully applied in recent work of Di Nasso, where he observed that there is a perfect correspondence between ultrafilters and ele- ments of the nonstandard universe up to a natural notion of equivalence. One the one hand, this allows one to manipulate ultrafilters as nonstandard points, and to use ultrafilter methods to prove the existence of certain combinatorial configurations in the nonstandard universe. One the other hand, this gives an intuitive and direct way to infer, from the existence of certain ultrafilter configurations, the existence of corresponding standard combinatorial configuration via the fundamental principle of transfer in the nonstandard method. This perspective has successfully been applied by Di Nasso and co-authors to the study of partition regularity problems for Diophantine equations over the integers, providing in particular a far-reaching generalization of the classical theorem of Rado on partition regularity of systems of linear equations. Unlike Rado’s theorem, this recent generalization also includes equations that are not linear. Finally, it is worth mentioning that many other results in combinatorics can be seen, directly or indirectly, as applications of the nonstandard method. For instance, the groundbreaking work of Hrushovski and Breuillard–Green–Tao on approximate groups, although not originally presented in this way, admit a natural nonstandard treatment. The same applies to the work of Bergelson and Tao on recurrence in quasirandom groups. The goal of this present manuscript is to introduce the uninitiated reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory. In particular, no previous knowledge of nonstandard analysis will be assumed. Instead, we will provide a complete and self-contained introduction to the nonstandard method in the first part of this book. Novel to our introduction is a treatment of the topic of iterated hyperex- tensions, which is crucial for some applications and has thus far appeared only in specialized research articles. The intended audience for this book include researchers in combinatorics that desire to get acquainted with the nonstandard approach, as well as experts of nonstandard analysis who have been working in this or other areas of research. The list of applications of the nonstandard method to combinatorics and Ramsey theory presented here is quite extensive, including cornerstone results of Ramsey theory such as Ramsey’s theorem, Hindman’s theorem on sets of finite sums, the Hales–Jewett theorem on variable words, and Gowers’ theorem on FINk. It then proceeds with results on partition regularity of diophantine equations and with density problems in combinatorial number theory. A nonstandard treatment of the triangle removal lemma, the Szemeredi´ reg- ularity lemma, and of the already mentioned work of Hrushovski and Breuillard–Green–Tao on approximate groups conclude the book. We hope that such a complete list of examples will help the reader unfamiliar with the nonstandard method get a good grasp on how the approach works and can be applied. At the same time, we believe that collecting these results together, and providing a unified presentation and approach, will provide a useful reference for researchers in the field and will further stimulate the research in this currently very active area. Pisa, Italy Irvine, California Pasadena, California Notation and Conventions We set N := f1;2;3;:::g to denote the set of positive natural numbers and N0 := f0;1;2;3;:::g to denote the set of natural numbers. We use the following conventions for elements of particular sets: • m and n always range over N; • k and l always range over Z; ∗ • H;K;M, and N always range over infinite elements of N; ∗ • d and e always denote (small) positive real numbers, while e denotes a positive infinitesimal element of R; • Given any set S, we let a, b, and g denote arbitrary (possibly standard) elements of ∗S; Fin(X) = fF ⊆ X j X is finiteg. For any n, we write [n] := f1;:::;ng. Similarly, we write [N] := f1;:::;Ng. jA\Ij Given any nonempty finite set I and any set A, we write d(A;I) := jIj .