Canonical Ramsey Theory on Polish Spaces

Abstract

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.

On Borel maps, calibrated ${\ensuremath {\sigma }}$-ideals, and homogeneity

“…The compactum M Ď f pUq is disjoint from D, hence boundary in f pUq, and we have, cf. (6) in Section 2, L Ď q f U rMs, which completes the proof of the claim.…”

“…Then, conditions (2), (3), (4) are met. Now, (4) guarantees that (5) the sequence ph n q uniformly converges to a continuous function g : XÑ H, (6) g|C n " h n |C n for n P N. From (2), (6) and the fact that the Cantor sets gpC n q are pairwise disjoint, cf. (3), we infer that for any Borel set A Ď X, ν´A X ď n C n¯" ÿ n νpA X C n q " ÿ n µpgpA X C n qq " µ´gpA X ď n C n q¯.…”

A definable E 0 class containing no definable elements

“…Let f be Δ 1 1 ( p) in L; p ∈ L ∩ ω ω . By Shoenfield, it is true in L[x] that f : 2 ω → ω ω is still a Δ 1 1 ( p) reduction of E 0 to E. 9 Let y = f (x). Then the E-class…”

F. P. Ramsey: The Theory, the Myth, and the Mirror

“…And a quick look at the Zentralblatt's list of authors who have published papers classified as Other combinatorial number theory, code 11B75, provides a Who's Who of Ramsey theory, including Paul Erd} os, Ron Graham, and Endre Szemerédi, which illustrates the close connection between Ramsey theory and Combinatorial Number theory. Monograph titles on Ramsey theory range from Ramsey Theory on Integers [12] and Elemental Methods in Ergodic Ramsey Theory [15] to Canonical Ramsey Theory on Polish Spaces [14]. A survey article [22] on applications of Ramsey theory includes a bibliography of 269 items.…”