Elemental methods in ergodic Ramsey theory / Randall McCutcheon

Auteur: McCutcheon, Randall (1965-) - AuteurType de document: Livre numériqueCollection: Lecture notes in mathematics, (Online) ; 1722Langue: anglaisÉditeur: Berlin : Springer, 1999 ISBN: 9783540668091 Note: This book gives an overview over the main results of chromatic and density Ramsey theory in 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. (Zentralblatt) Sujets MSC: 05D10 Combinatorics -- Extremal combinatorics -- Ramsey theory
54H20 General topology -- Connections with other structures, applications -- Topological dynamics
28D05 Measure and integration -- Measure-theoretic ergodic theory -- Measure-preserving transformations
37A05 Dynamical systems and ergodic theory -- Ergodic theory -- Measure-preserving transformations
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This book gives an overview over the main results of chromatic and density Ramsey theory in 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. (Zentralblatt)

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