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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.
Inhaltsverzeichnis
Ramsey Theory Old and New.- 1. Ramsey Numbers.- 2. Transfinite Ramsey Theory.- 3. Chromatic Number.- 4. Classical Theorems.- 5. Other Classical Theorems.- 6. Structural Generalizations.- 7. Infinite Ramsey Theorem.- 8. Unprovability Results.- 9. Non-Standard Applications.- I. Classics.- Problems and Results on Graphs and Hypergraphs: Similarities and Differences.- Note on Canonical Partitions.- II. Numbers.- On Size Ramsey Number of Paths, Trees and Circuits. II.- On the Computational Complexity of Ramsey-Type Problems.- Constructive Ramsey Bounds and Intersection Theorems for Sets.- Ordinal Types in Ramsey Theory and Well-Partial-Ordering Theory.- III. Structural Theory.- Partite Construction and Ramsey Space Systems.- Graham-Rothschild Parameter Sets.- Shelah's Proof of the Hales-Jewett Theorem.- IV. Noncombinatorial Methods.- Partitioning Topological Spaces.- Topological Ramsey Theory.- Ergodic Theory and Configurations in Sets of Positive Density.- V. Variations and Applications.-Topics in Euclidean Ramsey Theory.- On Pisier Type Problems and Results (Combinatorial Applications to Number Theory).- Combinatorial Statements Independent of Arithmetic.- Boolean Complexity and Ramsey Theorems.- Uncrowded Graphs.- Author Index.
