Read This! The MAA Online book review column Ramsey Theory on the Integers by Bruce M. Landman and Aaron Robertson Reviewed by Mihaela Poplicher

Ramsey Theory on the Integers: what a wonderful book! It would be great to be an undergraduate passionate about mathematics and come to read this book, as it contains a very "student-friendly" approach to one of the richest areas of mathematical research, Ramsey Theory.

As the authors mention at the end of their Preface, "this book would not exist without the essential contributions of the late Paul Erdös... Our professional lives would have had far less meaning and fulfillment without his work and his presence in our field. For that pervasive, though perhaps indirect, contribution to this text, we are in his debt."

The book is a very good way of introducing the students to mathematical research because it focuses on the Ramsey Theory on the set of integers, thus leaving out the rather complicated and hard to follow results in the general Ramsey Theory. The accessibility of the book does not make it uninteresting for mathematicians, as the many research problems included in each chapter (many unsolved problems for many years) constitute a very good resource of combinatorial or number theoretical problems, especially "Erdös-type" problems.

The book has an extensive bibliography: a list of 275 titles! This is to be expected, as Ramsey Theory is a very rich and popular area of mathematical research. However, there is no other book on the subject that refers only to Ramsey Theory on the set of integers and which is structured as a textbook for undergraduates. Even more, the background needed to read this book is just an elementary linear algebra course and a one semester course in abstract algebra.

The book can be used in a variety of ways, either as a textbook for a course, or as a source of research problems, as stated by the authors themselves:

• "as an undergraduate or graduate textbook for a second course in combinatorics or number theory;

• in an undergraduate or graduate seminar, a capstone course for undergraduates, or an independent study course;

• by students working under a REU program, or who are engaged in some other type of research experience;

• by graduate students looking for potential thesis topics;

• by the established researcher seeking a worthwhile resource in its material, its list of open research problems, and its somewhat enormous (often a fitting word when discussing Ramsey Theory) bibliography."

Just to give readers a better idea of the topics included in the book, as an extra argument in favor of its usefulness and beauty, here are some of the chapters:

• Preliminaries (beginning, of course, with the Pigeonhole Principle and Ramsey's Theorem);
• Van der Waerden's Theorem;
• Supersets of AP;
• Subsets of AP;
• Arithmetic Progressions (mod m);
• Schur's Theorem;
• Rado's theorem;
• Other Topics (including the Folkman-Rado-Sanders Theorem, as well as Patterns in Colorings)

Conclusion: I would strongly recommend this book for all researchers in Ramsey Theory, especially those looking to attract students to mathematical research. It really is a very good book: interesting, accessible and beautifully written. The authors really did a great job!

Publication Data: Ramsey Theory on the Integers, by Bruce M. Landman and Aaron Robertson. Student Mathematical Library, vol. 24. American Mathematical Society, 2003. Softcover, 320 pp., $49.00 ($39.00 to AMS members). ISBN: 0-8218-3199-2.

This page contains more information about the book, including a link to an errata file.

Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.

Posted August 9, 2004

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