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Ramsey Theory: Unsolved Problems and Results

Xiaodong Xu, Meilian Liang, and Haipeng Luo
Walter de Gruyter
Publication Date: 
Number of Pages: 
[Reviewed by
Matthew Katz
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In 1930, Frank P. Ramsey proved the result today known as Ramsey's theorem, but it was Pál Erdös who, starting in 1935, popularized Ramsey's theorem and wrote dozens of papers about its applications in many fields.  In the years since questions in Ramsey Theory have been difficult to answer; the tools used to answer one question are often tough to adapt for other questions.
In Ramsey Theory: Unsolved Problems and Results, the authors lay out and remark on dozens of open problems and conjectures in Ramsey theory.  The stated goal of the book is to look for unifying threads in these problems.  Problems are grouped together in chapters based on common themes or ways in which these problems might be attacked; some problems are addressed in more than one chapter to stress the different ways in which these problems can be viewed.  Many times the authors give some insight into the limitations of previously attempted methods. 
Besides the more standard applications of Ramsey theory to graph theory, there are also sections towards the end of the book devoted to questions related to van der Waerden's theorem, Szemeredi's Theorem, and other Ramsey-type theorems.
The book makes a great reference for the current state of results and related conjectures.  There no other books that come to mind which try to lay out and organize such a breadth of open problems in Ramsey theory as this one.  However, this is not a good book for someone who is just starting to learn Ramsey Theory.  Many chapters begin without motivation and just jump into defining a new object and then stating open problems.  Results that are given are almost always given without proof but usually with a reference of where the proof can be found.  The book also has some issues with grammar and formatting, but the more serious issue is the lack of citations for some statements.
I would recommend this book for anyone who is already familiar with Ramsey theory and looking for open problems in the field.
Matthew Katz is currently a teaching professor of mathematics at the Pennsylvania State University, where he got his Ph.D studying the theory of integer partitions under George Andrews in 2013.  His interests also include graph theory, number theory, and the history of mathematics.

See the table of contents in the publisher's webpage.