scholarly journals A RAMSEY THEOREM ON SEMIGROUPS AND A GENERAL VAN DER CORPUT LEMMA

2016 ◽  
Vol 81 (2) ◽  
pp. 718-741 ◽  
Author(s):  
ANUSH TSERUNYAN
Keyword(s):  
Ramsey Theory ◽  
Major Theme ◽  
Multiple Recurrence ◽  
Ramsey Theorem ◽  
Arithmetic Combinatorics

AbstractA major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemerédi’s theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs is that, for actions of semigroups, a certain kind of one recurrence (mixing along a filter) amplifies itself to multiple recurrence. This amplification is proved using a so-called van der Corput difference lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete filters) have been proven before (by Furstenberg, Bergelson–McCutcheon, and others), with a somewhat different proof in each case. We define a notion of differentiation for subsets of semigroups and isolate the class of filters that respect this notion. The filters in this class (call them ∂-filters) include all those for which the van der Corput lemma was known, and our main result is a van der Corput lemma for ∂-filters, which thus generalizes all its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup with edges between the semigroup elements labeled by their ratios.

scholarly journals Comparisons of Polychromatic and Monochromatic Ramsey Theory

2013 ◽  
Vol 78 (3) ◽  
pp. 951-968 ◽  
Author(s):  
Justin Palumbo
Keyword(s):  
Classical Result ◽  
Ramsey Theory ◽  
Axiom Of Choice ◽  
Ramsey Theorem ◽  
Partition Relations ◽  
Combinatorial Principles ◽  
The Axiom Of Choice ◽  
Relationship Of ◽  
The Relationship ◽  
Special Classes

AbstractWe compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

A Ramsey Theorem with an Application to Sequences in Banach Spaces

2012 ◽  
Vol 55 (2) ◽  
pp. 410-417
Author(s):  
Robert Service
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Ramsey Theory ◽  
Basic Sequence ◽  
Alternative Proof ◽  
Ramsey Theorem ◽  
Combinatorial Lemma

AbstractThe notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

scholarly journals A Dual Ramsey Theorem for Permutations

10.37236/6845 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Dragan Mašulović
Keyword(s):  
Category Theory ◽  
Ramsey Theory ◽  
Ramsey Property ◽  
Ramsey Theorem

In 2012 M. Sokić proved that the that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić.Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permutations has the dual Ramsey property as well. It was Leeb who pointed out in 1970 that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. In this paper we argue that this is even more the case when dealing with the dual Ramsey property.

scholarly journals MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SŁAWOMIR SOLECKI
Keyword(s):  
Ramsey Theory ◽  
Algebraic Structures ◽  
Ramsey Theorem

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem.

SUSTAINING FAMILY ROUTINES AFTER TRANSITIONING INTO PARENTHOOD: COUPLES’ PERCEPTIONS OF ASSISTING FACTORS

2016 ◽  
Vol 28 (1) ◽  
pp. 3-17
Author(s):  
Christine De Goede ◽  
Abraham P Greeff
Keyword(s):  
Low Income ◽  
Transition To Parenthood ◽  
Specific Problem ◽  
Family Members ◽  
Major Theme ◽  
Structured Interviews ◽  
Theory Analysis ◽  
Family Routines ◽  
Temporal Complexity ◽  
Context Specific

The aim of this qualitative study was to explore what assists couples in sustaining family routines after the transition to parenthood. Participants were recruited from two day-care centres in Cape Town, South Africa. In-depth, semi-structured interviews were conducted with 10 couples, mostly from low-income households, who had gone through this transition between one and four years previously. Grounded theory analysis revealed one major theme, Factors that decrease task and temporal complexity, with seven subthemes: Support from the wider family network; Couple cooperation and tag-teaming; Planning and pre-empting future problems; Adhering to schedules; Facilitative characteristics and skills of individual family members; Parents’ sense of commitment and responsibility towards family members; and idiosyncratic accommodations. Results underscore the need for professionals to help parents gain support from relatives; strengthen partner teamwork; foster schedule consistency; improve skills such as planning; foster their caretaker self-concepts; and facilitate context-specific problem-solving.

The Quranic Jesus: Prophet and Scapegoat

2019 ◽  
Vol 24 (1) ◽  
pp. 183-220
Author(s):  
John Ranieri
Keyword(s):  
Major Theme ◽  
The Bible ◽  
Christian Understanding ◽  
The Way

A major theme in René Girard’s work involves the role of the Bible in exposing the scapegoating practices at the basis of culture. The God of the Bible is understood to be a God who takes the side of victims. The God of the Qur’an is also a defender of victims, an idea that recurs throughout the text in the stories of messengers and prophets. In a number of ways, Jesus is unique among the prophets mentioned in the Qur’an. It is argued here that while the Quranic Jesus is distinctly Islamic, and not a Christian derivative, he functions in the Qur’an in a way analogous to the role Jesus plays in the gospels. In its depiction of Jesus, the Qur’an is acutely aware of mimetic rivalry, scapegoating, and the God who comes to the aid of the persecuted. Despite the significant differences between the Christian understanding of Jesus as savior and the way he is understood in the Qur’an, a Girardian interpretation of the Qur’anic Jesus will suggest ways in which Jesus can be a bridge rather than an obstacle in Christian/Muslim dialogue.

Systemic Hybridity in the Mediation of the American Presidential Campaign

2017 ◽  
Author(s):  
Andrew Chadwick
Keyword(s):  
Data Gathering ◽  
Real Space ◽  
Presidential Campaign ◽  
Newspaper Coverage ◽  
The Internet ◽  
Major Theme ◽  
Power Resources ◽  
Electoral Outcomes ◽  
Media System ◽  
Common Purpose

Chapter 7 continues the revisionist approach of chapter 6, but paints the 2008 U.S. presidential campaign on a broader canvas. Through a detailed analysis of key episodes in the mediation of the campaign, the chapter shows how the real-space spectacles of candidate appearances continue to generate the important television, radio, and newspaper coverage that remains so crucial for projecting the power of a candidate and conveying enthusiasm, movement, authenticity, and common purpose to both activists and nonactivists alike. The chapter discusses how these television-fuelled spectacles now also integrate with newer media logics of data-gathering, online fundraising, tracking, monitoring, and managed volunteerism. A major theme running through this chapter is the growing systemic integration of the internet and television in presidential campaigns. It also shows how the hybrid media system can shape electoral outcomes by providing new power resources for campaigns that can create and master the system's modalities.

The Rapture in American Imagination

2020 ◽  
pp. 261-275
Author(s):  
Tina Pippin
Keyword(s):  
Popular Culture ◽  
Jesus Christ ◽  
Major Theme ◽  
Political Climate ◽  
Historical Events ◽  
American Popular Culture ◽  
Left Behind ◽  
Television Film ◽  
Second Coming

The Rapture is the sudden and hoped for event of the second coming of Jesus Christ in the clouds to raise true Christian believers to heaven. American popular culture has played with this scenario in a variety of genres (e.g., television, film, novels), most often in a satirical way. From the faith perspective, the Rapture is a major theme in Christian fiction (the Left Behind series) and follows a timeline of political and historical events. Representations of the Rapture in popular culture often reflect the current political climate and the psychological anxiety, isolation, and sense of persecution of believers.

scholarly journals Recent results in partition (Ramsey) theory for finite lattices

1981 ◽  
Vol 35 (1-3) ◽  
pp. 185-198 ◽  
Author(s):  
Hans Jürgen Prömel ◽  
Bernd Voigt
Keyword(s):  
Ramsey Theory ◽  
Finite Lattices
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