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.
Vladimir Kanovei is a mathematician working in descriptive set theory and non-standard analysis. He is Leading Researcher in the Institute for Information Transmission Problems (IITP), Moscow, and Professor at Moscow State University of Railway Engineering (MIIT). Marcin Sabok works in mathematical logic and, in particular, in descriptive set theory. He received his PhD from Wroclaw University, Poland in 2009 and has held postdoctoral positions at the Kurt Godel Research Center for Mathematical Logic, Vienna and at the University of Illinois, Urbana-Champaign. Sabok is currently Assistant Professor in the Institute of Mathematics at the Polish Academy of Sciences. Jindrich Zapletal is a mathematician working in set theory. He developed a robust connection between descriptive set theory, abstract analysis and Shelah's method of proper forcing. His main research contributions are collected in the book Forcing Idealized (Cambridge University Press, 2008). Zapletal is Professor of Mathematics at the University of Florida.
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.