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