Teoria ergódica de Ramsey e a conjectura B + B + t

Abstract

In this work, we explore the relation between two distinct areas of Mathematics, namely Ergodic Theory and Ramsey Theory. The interaction between these two areas is nowadays called Ergodic Ramsey Theory. We will present the proof of two classical results: Roth theorem, where we will use tools from ergodic theory and spectral theory, and Hindman theorem, where we will use ultrafilters from set theory. Finally, we study the proof of Erdős B + B + t Conjecture, that was recently obtained using Ergodic Ramsey Theory. An element of great importance in this dissertation, that will be used both in the proof of Roth theorem and in the proof of B + B + t Conjecture, is the Kronecker factor, that captures an structure of rotation group inside a system that preserves measure.