Canonical Ramsey theory on Polish spaces (Q2849822)

From the back cover of the book: ``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.NEWLINENEWLINENEWLINEGiven 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.''NEWLINENEWLINENEWLINEThe book consists of ten chapters. The clearly written Introduction (Chapter 1) provides motivation (with basic results by Ramsey, Erdős and Rado (1950), Mycielski (1964), Silver (1970 and 1980), Prömel and Voigt (1985) and Mathias (1977), and Connes, Feldman and Weiss (1981)), basic concepts concerning sigma-ideals on Polish spaces like total canonization and Silver properties for a class of equivalence relations, free set property, mutual generics property, existence of square and rectangular coding functions as well as some properties of (analytic) equivalence relations, outline of results and navigation through the book.NEWLINENEWLINENEWLINE The first five chapters provide an introduction into the developed theory. Chapter 2 contains an introduction to descriptive set theory, forcing, generic ultrapowers, idealized forcing, and Katetov order and coding functions. Chapter 3 is devoted to analytic equivalence relations and models of set theory. Chapter 4 deals with classes of equivalence relations (smooth, countable, classifiable by countable structures, hypersmooth, analytic vs. Borel). Chapter 5 discusses games and Silver properties.NEWLINENEWLINENEWLINE The main new results appear in the last five chapters. They are devoted to different types of sigma-ideals (the game ideals, Ramsey-type ideals, product-type ideals, the countable support iteration ideals). For a good number of sigma-ideals, the authors prove the strongest canonization results possible. For example, in the case where \(I\) is a sigma-ideal on a compact space, sigma-generated by a coanalytic family of compact sets, they prove that if \(I\) is calibrated, then it has the free set property, total canonization for analytic equivalence relations, and the Silver property. In most cases such a strong canonization result is not available or the authors do not know how to prove it. In situations where the total canonization fails but obstacles are clearly identifiable, the authors provide a canonization up to the known obstacles (Theorems 9.3, 7.1, 7.7, 7.36). The canonization techniques are applied to achieve a number of ergodicity results for certain classical Borel equivalence relations (Theorems 6.66, 6.24, 6.67). The weakest canonization results obtained reduce the Borel reducibility complexity of equivalence relations to a special class (Theorems 9.26, 9.27, 8.17). There is also a number of anticanonization results. The last results have no counterpart in the finite or countable realm. Many results of the book either connect canonization properties of sigma-ideals with some forcing properties or use that connection in their proofs. Of special meaning is the connection between analytic equivalence relations and intermediate forcing extensions (Chapter 3).