Canonical Ramsey Theory on Polish Spaces

This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy-Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.Preface; 1. Introduction; 2. Background facts; 3. Analytic equivalence relations and models of set theory; 4. Classes of equivalence relations; 5. Games and the Silver property; 6. The game ideals; 7. Benchmark equivalence relations; 8. Ramsey-type ideals; 9. Product-type ideals; 10. The countable support iteration ideals; References; Index.