. Let E be a countable Borel equivalence relation on the space E ∞ of all inﬁnite partitions of the natural numbers. We show that E coincides with equality below a Carlson-Simpson generic element of E ∞ . In contrast, we show that there is a hypersmooth equivalence relation on E ∞ which is Borel bireducible with E 1 on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.

We prove a canonization result for the Carlson-Simpson forcing in the spirit of \cite{KSZ}. We generalize the weak form of the Carlson-Simpson theorem (\cite{CaSi}) dealing with partitions without… Expand

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not… Expand

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which… Expand

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.… Expand

This paper is a contribution to the study of Borel equivalence relations on standard
Borel spaces (i.e., Polish spaces equipped with their Borel structure). In
mathematics one often deals with… Expand

The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.Expand