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 Gödel 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. Jindřich 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.

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• 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.