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$α$-minimal Banach spaces

A Banach space with a Schauder basis is said to be $α$-minimal for some countable ordinal $α$ if, for any two block subspaces, the Bourgain embeddability index of one into the other 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.