Ramsey Theory for Discrete Structures

  • Hans Jürgen Prömel

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Roots of Ramsey Theory

    1. Front Matter
      Pages 1-1
    2. Hans Jürgen Prömel
      Pages 3-8
    3. Hans Jürgen Prömel
      Pages 9-29
  3. A Starting Point of Ramsey Theory: Parameter Sets

    1. Front Matter
      Pages 31-31
    2. Hans Jürgen Prömel
      Pages 33-39
    3. Hans Jürgen Prömel
      Pages 41-51
    4. Hans Jürgen Prömel
      Pages 53-59
    5. Hans Jürgen Prömel
      Pages 61-77
  4. Back to the Roots: Sets

    1. Front Matter
      Pages 79-79
    2. Hans Jürgen Prömel
      Pages 81-95
    3. Hans Jürgen Prömel
      Pages 97-103
    4. Hans Jürgen Prömel
      Pages 105-110
    5. Hans Jürgen Prömel
      Pages 111-118
    6. Hans Jürgen Prömel
      Pages 119-125
  5. Graphs and Hypergraphs

    1. Front Matter
      Pages 127-127
    2. Hans Jürgen Prömel
      Pages 129-144
    3. Hans Jürgen Prömel
      Pages 145-152
    4. Hans Jürgen Prömel
      Pages 153-169
    5. Hans Jürgen Prömel
      Pages 171-183

About this book

Introduction

This monograph covers some of the most important developments in Ramsey theory from its beginnings in the early 20th century via its many breakthroughs to recent important developments in the early 21st century.

 

The book first presents a detailed discussion of the roots of Ramsey theory before offering a thorough discussion of the role of parameter sets. It presents several examples of structures that can be interpreted in terms of parameter sets and features the most fundamental Ramsey-type results for parameter sets: Hales-Jewett's theorem and Graham-Rothschild¹s Ramsey theorem as well as their canonical versions and several applications. Next, the book steps back to the most basic structure, to sets. It reviews classic results as well as recent progress on Ramsey numbers and the asymptotic behavior of classical Ramsey functions. In addition, the chapter presents product versions of Ramsey's theorem, a combinatorial proof of the incompleteness of Peano arithmetic, provides a digression to discrepancy theory, and examines extensions of Ramsey's theorem to larger cardinals. The next chapter features an in-depth treatment of the Ramsey problem for graphs and hypergraphs. It gives an account on the existence of sparse and restricted Ramsey theorem's using sophisticated constructions as well as probabilistic methods. Among others it contains a proof of the induced Graham-Rothschild theorem and the random Ramsey theorem. The book closes with a chapter on one of the recent highlights of Ramsey theory: a combinatorial proof of  the density Hales-Jewett theorem.

 

This book provides graduate students as well as advanced researchers with a solid introduction and reference to the field.

 

Keywords

Ramsey theory combinatorics discrete structures

Authors and affiliations

  • Hans Jürgen Prömel
    • 1
  1. 1.Technische Universität DarmstadtDarmstadtGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-01315-2
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-01314-5
  • Online ISBN 978-3-319-01315-2
  • About this book