Before the 1990's, applications of Ramsey theory already existed in Banach space theory. Many of these involved using the Galvin-Prikry theorem to stabilize a property relative to the subsequences of a basic sequence. However, no Ramsey-type theorems for the block subsequences of a basic sequence were known. In fact, many such statements turned out to be false. In the early 90's, Gowers found a correct formulation of a block Ramsey principle in which one half of the dichotomy postulates the existence of a winning strategy for a subspace/vector game. Gowers and others then used this principle to solve several problems in the field which had been open since Banach's time. In 1998, Gowers received the Fields Medal for his work connecting infinitary combinatorics and functional analysis.

In this talk, we will explain Gowers' game and how it leads to the notion of a collection of block basic sequences being "weakly Ramsey." We will then define a Polish topology on the collection of block basic sequences such that all analytic sets are weakly Ramsey. Only the proof for open collections will be presented.