Ramsey Theory for Discrete Structures

ISBN-13: 9783319013145 / Angielski / Twarda / 2013 / 232 str.

Hans Jurgen Promel

Opis

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-Rothschilds 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, it 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 part of the book 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.

Recenzje

From the book reviews:

"The results and proofs covered in the book are clearly explained and the material covered is extensive. I therefore think this book provides a strong introductory reference for students. ... Further, the monograph will be a great addition to the bookshelves of researchers in the field of Ramsey theory and related areas, as a reference on both classical results and some of the more recent breakthroughs in the subject." (Andrew Clark Treglown, Mathematical Reviews, October, 2014)

"Prömel (Technical Univ. Darmstadt, Germany) includes three gems of this episodic subject, each sufficient to sell the book: a self-contained treatment of the Paris-Harrington theorem showing that a certain Ramsey-type theorem witnesses the incompleteness of Peano arithmetic; a detailed treatment of the Graham-Rothschild theorem long ago made famous for its fast-growing functions by M. Gardner's Scientific American column; and a very short proof (now) of Szemerédi's theorem ... . Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 51 (11), August, 2014)

Spis treści

Foreword by Angelika Steger.- Preface.- Conventions.- Part I Roots of Ramsey Theory: 1.1 Ramsey’s theorem.- 1.2 From Hilbert’s cube lemma to Rado’s thesis.- Part II A Starting Point of Ramsey Theory: Parameter Sets: 2.1 Definitions and basic examples.- 2.2 Hales-Jewett’s theorem.- 2.3 Graham-Rothschild’s theorem.- 2.4 Canonical partitions.- Part III Back to the Roots: Sets: 3.1 Ramsey numbers.- 3.2 Rapidly growing Ramsey functions.- 3.3 Product theorems.- 3.4 A quasi Ramsey theorem.- 3.5 Partition relations for cardinal numbers.- Part IV Graphs and Hypergraphs: 4.1 Finite graphs.- 4.2 Infinite graphs.- 4.3 Hypergraphs on parameter sets.- 4.4. Ramsey statements for random graphs.- 4.5 Sparse Ramsey Theorems.- Part V Density Ramsey Theorems: 5.1 Szemerédi’s Theorem.- 5.2 Density Hales-Jewett Theorem.- 5.3 Proof of the density Hales-Jewett theorem.- References.- Index.

O autorze

The author was one of the driving forces of the field in the eighties and nineties of the last century. Many of the, by now standard, proofs of the field were discovered by him.

Blurb

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.