Elemental methods in ergodic Ramsey theory
This book gives an overview over the main results of chromatic and density Ramsey theory in \(\mathbb{Z}\). It is based on lecture notes for a graduate course and therefore additionally provides a large variety of exercises and problems that are helpful for a better understanding of the field. Even though many of the presented results were originally proven combinatorially, most of the proofs in this book are based on topological dynamics and ergodic theory. The problems considered mainly fall into the classes of van der Waerden type theorems, Hindman type theorems and Szemerédi type theorems. After an introduction to Ramsey theory and to topological dynamics in the first chapter, results from infinitary Ramsey theory and density Ramsey theory are presented in Chapters 2 and 3. In Chapters 4 and 5 ergodic theory is used to establish double recurrence results and multiple recurrence results. In particular, Roth theorems and Szeméredi theorems are proven in this context.
scientific article; zbMATH DE number 1391017