Ramsey theory is a dynamic area of combinatorics that has various applications in analysis, ergodic theory, logic, number theory, probability theory, theoretical computer science, and topological dynamics. This book is devoted to one of the most important areas of Ramsey theory--the Ramsey theory of product spaces. It is a culmination of a series of recent breakthroughs by the two authors and their students who were able to lift this theory to the infinite-dimensional case. The book presents many major results and methods in the area, such as Szemeredi’s regularity method, the hypergraph removal lemma, and the density Hales-Jewett theorem. This book addresses researchers in combinatorics but also working mathematicians and advanced graduate students who are interested in Ramsey theory. The prerequisites for reading this book are rather minimal: it only requires familiarity, at the graduate level, with probability theory and real analysis. Some familiarity with the basics of Ramsey theory would be beneficial, though not necessary. I think that this book has a good chance of becoming a classic on density Ramsey theory at the level of the Graham-Rothschild-Spencer book on basic Ramsey theory. --Stevo Todorcevic, University of Toronto The book by Dodos and Kanellopoulos is first-rate! It is timely, well written, and has a great selection of topics. --Ron Graham, University of California, San Diego

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