• Gowers
• Ramsey theory
• combinatorics
• ultrafilter

In this thesis we give an overview of Gowers' combinatorial results for the set of maps $\text{FIN}_{k}^{\pm}$, and their generalisations. In the first chapter we introduce the theory of ultrafilters, which are our fundamental tool, and the basic notions about semigroups. The second chapter is the core of this work, here we present the notion of subsymmetric ultrafilters and we will use them to prove Gowers' theorems. In the third chapter, we will see briefly Gowers' original arguments. Finally, in the fourth chapter, we will see M. Lupini's recent results on generalised tetris operations.