Ebook: Mathematics of Ramsey Theory

One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.

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One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.

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One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.
Content:
Front Matter....Pages I-XIV
Introduction Ramsey Theory Old and New....Pages 1-9
Front Matter....Pages 11-11
Problems and Results on Graphs and Hypergraphs: Similarities and Differences....Pages 12-28
Note on Canonical Partitions....Pages 29-32
Front Matter....Pages 33-33
On Size Ramsey Number of Paths, Trees and Circuits. II....Pages 34-45
On the Computational Complexity of Ramsey—Type Problems....Pages 46-52
Constructive Ramsey Bounds and Intersection Theorems for Sets....Pages 53-56
Ordinal Types in Ramsey Theory and Well-Partial-Ordering Theory....Pages 57-95
Front Matter....Pages 97-97
Partite Construction and Ramsey Space Systems....Pages 98-112
Graham-Rothschild Parameter Sets....Pages 113-149
Shelah’s Proof of the Hales-Jewett Theorem....Pages 150-151
Front Matter....Pages 153-153
Partitioning Topological Spaces....Pages 154-171
Topological Ramsey Theory....Pages 172-183
Ergodic Theory and Configurations in Sets of Positive Density....Pages 184-198
Front Matter....Pages 199-199
Topics in Euclidean Ramsey Theory....Pages 200-213
On Pisier Type Problems and Results (Combinatorial Applications to Number Theory)....Pages 214-231
Combinatorial Statements Independent of Arithmetic....Pages 232-245
Boolean Complexity and Ramsey Theorems....Pages 246-252
Uncrowded Graphs....Pages 253-262
Back Matter....Pages 263-272

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One of the important areas of contemporary combinatorics is Ramsey theory. Ramsey theory is basically the study of structure preserved under partitions. The general philosophy is reflected by its interdisciplinary character. The ideas of Ramsey theory are shared by logicians, set theorists and combinatorists, and have been successfully applied in other branches of mathematics. The whole subject is quickly developing and has some new and unexpected applications in areas as remote as functional analysis and theoretical computer science. This book is a homogeneous collection of research and survey articles by leading specialists. It surveys recent activity in this diverse subject and brings the reader up to the boundary of present knowledge. It covers virtually all main approaches to the subject and suggests various problems for individual research.
Content:
Front Matter....Pages I-XIV
Introduction Ramsey Theory Old and New....Pages 1-9
Front Matter....Pages 11-11
Problems and Results on Graphs and Hypergraphs: Similarities and Differences....Pages 12-28
Note on Canonical Partitions....Pages 29-32
Front Matter....Pages 33-33
On Size Ramsey Number of Paths, Trees and Circuits. II....Pages 34-45
On the Computational Complexity of Ramsey—Type Problems....Pages 46-52
Constructive Ramsey Bounds and Intersection Theorems for Sets....Pages 53-56
Ordinal Types in Ramsey Theory and Well-Partial-Ordering Theory....Pages 57-95
Front Matter....Pages 97-97
Partite Construction and Ramsey Space Systems....Pages 98-112
Graham-Rothschild Parameter Sets....Pages 113-149
Shelah’s Proof of the Hales-Jewett Theorem....Pages 150-151
Front Matter....Pages 153-153
Partitioning Topological Spaces....Pages 154-171
Topological Ramsey Theory....Pages 172-183
Ergodic Theory and Configurations in Sets of Positive Density....Pages 184-198
Front Matter....Pages 199-199
Topics in Euclidean Ramsey Theory....Pages 200-213
On Pisier Type Problems and Results (Combinatorial Applications to Number Theory)....Pages 214-231
Combinatorial Statements Independent of Arithmetic....Pages 232-245
Boolean Complexity and Ramsey Theorems....Pages 246-252
Uncrowded Graphs....Pages 253-262
Back Matter....Pages 263-272
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