Canonical Ramsey Theory on Polish Spaces

Vladimir Kanovei, Marcin Sabok, Jindřich Zapletal

• Publisher: Cambridge University Press
• Year: 2013
• Print ISBN: 9781107026858
• eBook ISBN: 9781107439535
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

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.

Table of contents

• Cover
• Contents
• 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