Softcover ISBN:  9780821898673 
Product Code:  STML/73 
384 pp 
List Price:  $61.00 
Individual Price:  $48.80 
Electronic ISBN:  9781470420000 
Product Code:  STML/73.E 
384 pp 
List Price:  $61.00 
Individual Price:  $48.80 

Book DetailsStudent Mathematical LibraryVolume: 73; 2014MSC: Primary 05;
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 ErdősGinzbergZiv 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.Reviews of the Previous Edition:
Students will enjoy it due to the highly accessible exposition of the material provided by the authors.
—MAA Horizons
What a wonderful book! … contains a very “student friendly” approach to one of the richest areas of mathematical research … a very good way of introducing the students to mathematical research … an extensive bibliography … no other book on the subject … which is structured as a textbook for undergraduates … The book can be used in a variety of ways, either as a textbook for a course, or as a source of research problems … strongly recommend this book for all researchers in Ramsey theory … very good book: interesting, accessible and beautifully written. The authors really did a great job!
—MAA Online
ReadershipUndergraduate and graduate students interested in combinatorics, number theory, and Ramsey theory.

Table of Contents

Chapters

Chapter 1. Preliminaries

Chapter 2. Van der Waerden’s theorem

Chapter 3. Supersets of $AP$

Chapter 4. Subsets of $AP$

Chapter 5. Other generalizations of $w(k;r)$

Chapter 6. Arithmetic progressions $(\mathrm {mod}\,m)$

Chapter 7. Other variations on van der Waerden’s theorem

Chapter 8. Schur’s theorem

Chapter 9. Rado’s theorem

Chapter 10. Other topics


Additional Material

Reviews

This is an excellent undergraduate text which provides students with an introduction to research; it is also a source for all those who are interested in combinatorial or number theoretic problems. ... The textbook is carefully written. I recommend it to students interested in combinatorics and to their teachers as well.
Monatshafte für Mathematik


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 Get Permissions
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 ErdősGinzbergZiv 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.
Reviews of the Previous Edition:
Students will enjoy it due to the highly accessible exposition of the material provided by the authors.
—MAA Horizons
What a wonderful book! … contains a very “student friendly” approach to one of the richest areas of mathematical research … a very good way of introducing the students to mathematical research … an extensive bibliography … no other book on the subject … which is structured as a textbook for undergraduates … The book can be used in a variety of ways, either as a textbook for a course, or as a source of research problems … strongly recommend this book for all researchers in Ramsey theory … very good book: interesting, accessible and beautifully written. The authors really did a great job!
—MAA Online
Undergraduate and graduate students interested in combinatorics, number theory, and Ramsey theory.

Chapters

Chapter 1. Preliminaries

Chapter 2. Van der Waerden’s theorem

Chapter 3. Supersets of $AP$

Chapter 4. Subsets of $AP$

Chapter 5. Other generalizations of $w(k;r)$

Chapter 6. Arithmetic progressions $(\mathrm {mod}\,m)$

Chapter 7. Other variations on van der Waerden’s theorem

Chapter 8. Schur’s theorem

Chapter 9. Rado’s theorem

Chapter 10. Other topics

This is an excellent undergraduate text which provides students with an introduction to research; it is also a source for all those who are interested in combinatorial or number theoretic problems. ... The textbook is carefully written. I recommend it to students interested in combinatorics and to their teachers as well.
Monatshafte für Mathematik