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.
General
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