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Ramsey Theory: Yesterday, Today, and Tomorrow

Alexander Soifer, editor
Publication Date: 
Number of Pages: 
Progress in Mathematics 285
[Reviewed by
Miklós Bóna
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As we learn from the preface, the book grew out of an intentionally non-traditional conference on Ramsey theory. In accordance with that, the book itself is far from being a traditional textbook or reference book on the subject.

There are several ways in which the book is unique. We learn far more about the history of Ramsey theory than from other sources. This is mostly due to the articles written by the editor, though the survey paper of Joel Spencer on determining the Ramsey number R(3,k) also covers 80 years.

Following the promise made in the title of the book, the present of Ramsey theory is represented by several traditional survey papers. We will not list these: see the table of contents.

Finally, the promise of discussing the future is fulfilled by a very extensive list of open problems contributed by numerous participants. Sometimes it is not even clear what the best way of asking a certain question is, and we are shown the raw form of the problem, just as we would if we participated at a conference.

The book suffers from some minor inconsistencies as most books with ten authors do. Various authors write the same notions, or even names, differently, letting the reader guess if these notions, or people, are indeed distinct. Hungarian names take a particularly vicious beating. That aside, the book is undoubtedly a lot of fun. As one expects from a book on Ramsey theory, it is full of problems that are very easy to formulate, but terribly hard to solve. Because of its non-traditional nature, the book will probably not be a frequent choice for a regular graduate course, but could well be used for a course using a seminar format, or for self-study by a graduate student familiarizing himself with the subject.

Miklós Bóna is Professor of Mathematics at the University of Florida.

How This Book Came into Being

Ramsey Theory before Ramsey, Prehistory and Early History: An Essay in 13 Parts
Alexander Soifer

Eighty Years of Ramsey R(3, k)… and Counting!
Joel Spencer

Ramsey Numbers Involving Cycles
Stanislaw P. Radziszowski

On the function of Erdős and Rogers
Andrzej Dudek and Voytĕch Rödl

Large Monochromatic Components in Edge Colorings of Graphs
András Gyárfás

Szlam’s Lemma: Mutant Offspring of a Euclidean Ramsey Problem: From 1973, with Numerous Applications
Jeffrey Burkert and Peter Johnson Jr.

Open Problems in Euclidean Ramsey Theory
Ron Graham and Eric Tressler

Chromatic Number of the Plane and Its Relatives, History, Problems and Results: An Essay in 11 Parts
Alexander Soifer

Euclidean Distance Graphs on the Rational Points
Peter Johnson Jr.

Open Problems Session