Canonical Ramsey Theory on Polish Spaces (Cambridge Tracts in Mathematics)

Canonical Ramsey Theory on Polish Spaces (Cambridge Tracts in Mathematics)

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Vladimir Kanovei, Marcin Sabok, Jindřich Zapletal
Cambridge University Press, 9/12/2013
EAN 9781107026858, ISBN10: 1107026857

Hardcover, 278 pages, 22.8 x 15.2 x 1.9 cm
Language: English

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.

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