α-Minimal Banach spaces

A Banach space W with a Schauder basis is said to be α-minimal for some α<ω1 if, for any two block subspaces Z,Y⊆W, the Bourgain embeddability index of Z into Y is at least α. We prove a dichotomy that characterises when a Banach space has an α-minimal subspace, which contributes to the ongoing project, initiated by W.T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.