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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

  • Book
  • © 2019

Overview

Authors:
  1. Mauro Di Nasso

    1. Department of Mathematics, Universita di Pisa, Pisa, Italy

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  2. Isaac Goldbring

    1. Department of Mathematics, University of California, Irvine, Irvine, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Martino Lupini

    1. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

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  • Presents the first introduction to nonstandard methods geared towards combinatorics
  • Includes new nonstandard proofs of classical results in Ramsey theory
  • Provides a self-contained introduction to the literature of nonstandard methods in combinatorics
  • Numerous exercises help readers to familiarize themselves with the material

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2239)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Preliminaries

    1. Front Matter

      Pages 1-1
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    2. Ultrafilters

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 3-10

    3. Nonstandard Analysis

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 11-43

    4. Hyperfinite Generators of Ultrafilters

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 45-48

    5. Many Stars: Iterated Nonstandard Extensions

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 49-54

    6. Loeb Measure

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 55-70

  3. Ramsey Theory

    1. Front Matter

      Pages 71-71
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    2. Ramsey’s Theorem

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 73-79

    3. The Theorems of van der Waerden and Hales-Jewett

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 81-88

    4. From Hindman to Gowers

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 89-98

    5. Partition Regularity of Equations

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 99-108

  4. Combinatorial Number Theory

    1. Front Matter

      Pages 109-109
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    2. Densities and Structural Properties

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 111-122

    3. Working in the Remote Realm

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 123-131

    4. Jin’s Sumset Theorem

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 133-143

    5. Sumset Configurations in Sets of Positive Density

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 145-152

    6. Near Arithmetic Progressions in Sparse Sets

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 153-160

    7. The Interval Measure Property

      • Mauro Di Nasso, Isaac Goldbring, Martino Lupini
      Pages 161-169

  5. Other Topics

    1. Front Matter

      Pages 171-171
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Keywords

About this book

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.


Authors and Affiliations

  • Department of Mathematics, Universita di Pisa, Pisa, Italy

    Mauro Di Nasso

  • Department of Mathematics, University of California, Irvine, Irvine, USA

    Isaac Goldbring

  • School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Martino Lupini

Bibliographic Information

  • Book Title: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

  • Authors: Mauro Di Nasso, Isaac Goldbring, Martino Lupini

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-17956-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Softcover ISBN: 978-3-030-17955-7Published: 24 May 2019

  • eBook ISBN: 978-3-030-17956-4Published: 23 May 2019

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XVI, 206

  • Number of Illustrations: 2 b/w illustrations

  • Topics: Combinatorics, Mathematical Logic and Foundations

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