Nonstandard Methods In Ramsey Theory And Combinatorial Number Theory

Author : Mauro Di Nasso,Isaac Goldbring,Martino Lupini
Publisher : Springer
Page : 206 pages
File Size : 53,7 Mb
Release : 2019-05-23
Category : Mathematics
ISBN : 9783030179564

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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory by Mauro Di Nasso,Isaac Goldbring,Martino Lupini Pdf

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 initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.

Author : Anonim
Publisher : Unknown
Page : 0 pages
File Size : 50,5 Mb
Release : 2020
Category : Combinatorial number theory
ISBN : 3030311074

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Combinatorial and Additive Number Theory III by Anonim Pdf

Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Author : Matthew Katz,Jan Reimann
Publisher : American Mathematical Soc.
Page : 207 pages
File Size : 43,8 Mb
Release : 2018-10-03
Category : Combinatorial analysis
ISBN : 9781470442903

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An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics by Matthew Katz,Jan Reimann Pdf

This book takes the reader on a journey through Ramsey theory, from graph theory and combinatorics to set theory to logic and metamathematics. Written in an informal style with few requisites, it develops two basic principles of Ramsey theory: many combinatorial properties persist under partitions, but to witness this persistence, one has to start with very large objects. The interplay between those two principles not only produces beautiful theorems but also touches the very foundations of mathematics. In the course of this book, the reader will learn about both aspects. Among the topics explored are Ramsey's theorem for graphs and hypergraphs, van der Waerden's theorem on arithmetic progressions, infinite ordinals and cardinals, fast growing functions, logic and provability, Gödel incompleteness, and the Paris-Harrington theorem. Quoting from the book, “There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time. Ramsey theory is one of those ladders.”

Author : Sukumar Das Adhikari
Publisher : Unknown
Page : 184 pages
File Size : 42,6 Mb
Release : 2002
Category : Combinatorial analysis
ISBN : 8173193037

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Aspects of Combinatorics and Combinatorial Number Theory by Sukumar Das Adhikari Pdf

Author : Melvyn B. Nathanson
Publisher : Springer Nature
Page : 237 pages
File Size : 51,5 Mb
Release : 2019-12-10
Category : Mathematics
ISBN : 9783030311063

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Combinatorial and Additive Number Theory III by Melvyn B. Nathanson Pdf

Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Author : Vieri Benci,Mauro Di Nasso
Publisher : World Scientific
Page : 346 pages
File Size : 48,5 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9789813276604

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How To Measure The Infinite: Mathematics With Infinite And Infinitesimal Numbers by Vieri Benci,Mauro Di Nasso Pdf

'This text shows that the study of the almost-forgotten, non-Archimedean mathematics deserves to be utilized more intently in a variety of fields within the larger domain of applied mathematics.'CHOICEThis book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the Alpha-Theory, which can be considered as an alternative approach to nonstandard analysis.The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory.The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid's Principle: 'The whole is larger than its parts'. The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory.

Author : Spiros A. Argyros,Stevo Todorcevic
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 47,7 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9783764373603

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Ramsey Methods in Analysis by Spiros A. Argyros,Stevo Todorcevic Pdf

This book contains two sets of notes prepared for the Advanced Course on R- sey Methods in Analysis given at the Centre de Recerca Matem` atica in January 2004, as part of its year-long research programme on Set Theory and its Appli- tions. The common goal of the two sets of notes is to help young mathematicians enter a very active area of research lying on the borderline between analysis and combinatorics. The solution of the distortion problem for the Hilbert space, the unconditional basic sequence problem for Banach spaces, and the Banach ho- geneous space problem are samples of the most important recent advances in this area, and our two sets of notes will give some account of this. But our main goal was to try to expose the general principles and methods that lie hidden behind and are most likely useful for further developments. The goal of the ?rst set of notes is to describe a general method of building norms with desired properties, a method that is clearly relevant when testing any sort of intuition about the in?nite-dimensional geometry of Banach spaces. The goal of the second set of notes is to expose Ramsey-theoretic methods relevant for describing the rough structure present in this sort of geometry. We would like to thank the coordinator of the Advanced Course, Joan Ba- ria, and the director of the CRM, Manuel Castellet, for giving us this challenging but rewarding opportunity. Part A SaturatedandConditional StructuresinBanachSpaces SpirosA.

Author : Xiaodong Xu,Meilian Liang,Haipeng Luo
Publisher : Walter de Gruyter GmbH & Co KG
Page : 190 pages
File Size : 46,9 Mb
Release : 2018-08-06
Category : Mathematics
ISBN : 9783110576634

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Ramsey Theory by Xiaodong Xu,Meilian Liang,Haipeng Luo Pdf

Key problems and conjectures have played an important role in promoting the development of Ramsey theory, a field where great progress has been made during the past two decades, with some old problems solved and many new problems proposed. The present book will be helpful to readers who wish to learn about interesting problems in Ramsey theory, to see how they are interconnected, and then to study them in depth. This book is the first problem book of such scope in Ramsey theory. Many unsolved problems, conjectures and related partial results in Ramsey theory are presented, in areas such as extremal graph theory, additive number theory, discrete geometry, functional analysis, algorithm design, and in other areas. Most presented problems are easy to understand, but they may be difficult to solve. They can be appreciated on many levels and by a wide readership, ranging from undergraduate students majoring in mathematics to research mathematicians. This collection is an essential reference for mathematicians working in combinatorics and number theory, as well as for computer scientists studying algorithms. Contents Some definitions and notations Ramsey theory Bi-color diagonal classical Ramsey numbers Paley graphs and lower bounds for R(k, k) Bi-color off-diagonal classical Ramsey numbers Multicolor classical Ramsey numbers Generalized Ramsey numbers Folkman numbers The Erdős–Hajnal conjecture Other Ramsey-type problems in graph theory On van der Waerden numbers and Szemeredi’s theorem More problems of Ramsey type in additive number theory Sidon–Ramsey numbers Games in Ramsey theory Local Ramsey theory Set-coloring Ramsey theory Other problems and conjectures

Author : Anonim
Publisher : Unknown
Page : 0 pages
File Size : 50,8 Mb
Release : 2020
Category : Combinatorial number theory
ISBN : 8303031104

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Combinatorial and Additive Number Theory III by Anonim Pdf

Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Author : Isaac Goldbring
Publisher : American Mathematical Society
Page : 421 pages
File Size : 42,6 Mb
Release : 2022-06-28
Category : Mathematics
ISBN : 9781470469610

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Ultrafilters Throughout Mathematics by Isaac Goldbring Pdf

Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty.

Author : Pandelis Dodos,Vassilis Kanellopoulos
Publisher : American Mathematical Soc.
Page : 245 pages
File Size : 53,8 Mb
Release : 2016-05-16
Category : Combinatorial analysis
ISBN : 9781470428082

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Ramsey Theory for Product Spaces by Pandelis Dodos,Vassilis Kanellopoulos Pdf

Ramsey theory is a dynamic area of combinatorics that has various applications in analysis, ergodic theory, logic, number theory, probability theory, theoretical computer science, and topological dynamics. This book is devoted to one of the most important areas of Ramsey theory—the Ramsey theory of product spaces. It is a culmination of a series of recent breakthroughs by the two authors and their students who were able to lift this theory to the infinite-dimensional case. The book presents many major results and methods in the area, such as Szemerédi's regularity method, the hypergraph removal lemma, and the density Hales–Jewett theorem. This book addresses researchers in combinatorics but also working mathematicians and advanced graduate students who are interested in Ramsey theory. The prerequisites for reading this book are rather minimal: it only requires familiarity, at the graduate level, with probability theory and real analysis. Some familiarity with the basics of Ramsey theory would be beneficial, though not necessary.

Author : Bruce M. Landman, Aaron Robertson
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 52,8 Mb
Release : 2014-11-10
Category : Mathematics
ISBN : 9780821898673

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Ramsey Theory on the Integers by Bruce M. Landman, Aaron Robertson Pdf

Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an "inequality" version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. Many new results and proofs have been added, most of which were not known when the first edition was published. Furthermore, the book's tables, exercises, lists of open research problems, and bibliography have all been significantly updated. This innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subject. This breakthrough book will engage students, teachers, and researchers alike.

Author : Bruce Landman,Melvyn B. Nathanson,Jaroslav Nešetril,Richard J. Nowakowski,Carl Pomerance,Aaron Robertson
Publisher : Walter de Gruyter
Page : 166 pages
File Size : 42,7 Mb
Release : 2013-08-29
Category : Mathematics
ISBN : 9783110280616

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Combinatorial Number Theory by Bruce Landman,Melvyn B. Nathanson,Jaroslav Nešetril,Richard J. Nowakowski,Carl Pomerance,Aaron Robertson Pdf

This volume contains selected refereed papers based on lectures presented at the "Integers Conference 2011", an international conference in combinatorial number theory that was held in Carrollton, Georgia, United States in October 2011. This was the fifth Integers Conference, held bi-annually since 2003. It featured plenary lectures presented by Ken Ono, Carla Savage, Laszlo Szekely, Frank Thorne, and Julia Wolf, along with sixty other research talks. This volume consists of ten refereed articles, which are expanded and revised versions of talks presented at the conference. They represent a broad range of topics in the areas of number theory and combinatorics including multiplicative number theory, additive number theory, game theory, Ramsey theory, enumerative combinatorics, elementary number theory, the theory of partitions, and integer sequences.

Author : Jiří Matousek,Jaroslav Nešetřil,Marco Pellegrini
Publisher : Springer
Page : 160 pages
File Size : 42,6 Mb
Release : 2015-04-09
Category : Mathematics
ISBN : 9788876425257

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Geometry, Structure and Randomness in Combinatorics by Jiří Matousek,Jaroslav Nešetřil,Marco Pellegrini Pdf

​This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include: graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.

Author : Vitaly Bergelson
Publisher : American Mathematical Soc.
Page : 214 pages
File Size : 42,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821848333

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Ultrafilters across Mathematics by Vitaly Bergelson Pdf

Presents the state-of-the-art of applications in the whole spectrum of mathematics which are grounded on the use of ultrafilters and ultraproducts. It contains two general surveys on ultrafilters in set theory and on the ultraproduct construction, as well as papers that cover additive and combinatorial number theory, nonstandard methods and stochastic differential equations, measure theory, dynamics, Ramsey theory, algebra in the space of ultrafilters, and large cardinals.