Definable sets containing productsets in expansions of groups

Andrews, Uri; Conant, Gabriel; Goldbring, Isaac

doi:10.1515/jgth-2018-0038

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…When this paper was essentially complete, we became aware of [2], which is worth mentioning since it employs similar methods in a related context.…”

Section: Introductionmentioning

confidence: 99%

Ramsey’s Coheirs

Colla¹,

Zambella²

2021

J. symb. log.

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We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey's theorem itself. Then we prove Hindman's theorem and the Hales-Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers. §1. Introduction. Ramsey theory has substantial and diverse applications to many parts of mathematics. In particular, Ramsey's theorem has foundational applications to model theory through the Ehrenfeucth-Mostowski construction of indiscernibles and generalizations thereof. In this paper we explore the converse direction, that is, we use model theory to obtain new proofs of classical results in Ramsey Theory.The Stone-Čech compactification, obtained via ultrafilters, is a widely employed method for proving Ramsey theoretic results. One of its first major applications is the celebrated Galvin-Glazer proof of Hindman's theorem, see, e.g., [4]. Our methods are related, but alternative, to the ultrafilter approach. We replace G (the Stone-Čech compactification of a semigroup G) with a large saturated elementary extension of G, i.e., a monster model of Th(G/G). One immediate advantage is that we work with elements of a natural semigroup with a natural operation. In contrast, elements of G are ultrafilters, that is, sets of sets, and the semigroup operation among ultrafilters is far from straightforward.This idea is not completely new: in his seminal work on the applications of topological dynamics to model theory [14, 15], Newelski replaces the semigroup G with the space of types over G with a suitably defined operation. Our approach is similar, except that, unlike Newelski, we do not pursue connections with topological dynamics, but rather offer an alternative realm of application. The investigation of alternative methods in the study of regularity phenomena has been called for by Di Nasso [5, Open problem #1]. This article contains a possible answer.The model theoretic tools employed in this paper are relatively basic. Section 2 is meant to give an accessible overview of the necessary notions for readers whose expertise is not primarily in model theory. Our results do not require assumptions of model theoretic tameness such as stability, NIP, etc., much like those that use

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“…When this paper was essentially complete, we became aware of [2], which is worth mentioning since it employs similar methods in a related context.…”

Section: Introductionmentioning

confidence: 99%

Ramsey’s Coheirs

Colla¹,

Zambella²

2021

J. symb. log.

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“…As a corollary, derived using Ramsey's theorem and a result of Hindman [Hin82, Theorem 3.8], it follows that if A has positive upper density, then for some t ∈ N the union A ∪ (A − t) contains a sum B + C where B and C are infinite sets. Some further progress on a variant of Conjecture 1.1 was also made in [ACG17].…”

Section: Introductionmentioning

confidence: 99%

A proof of a sumset conjecture of Erdős

2019

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In this paper we show that every set A ⊂ N with positive density contains B + C for some pair B, C of infinite subsets of N, settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.

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Definable sets containing productsets in expansions of groups

Cited by 2 publications

References 16 publications

Ramsey’s Coheirs

Ramsey’s Coheirs

A proof of a sumset conjecture of Erdős

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