This is an announcement for the paper "$\alpha$-minimal Banach spaces" by Christian Rosendal. Abstract: 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. Archive classification: math.FA math.LO Mathematics Subject Classification: 46B03, 03E15 Submitted from: rosendal@math.uic.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.3543 or http://arXiv.org/abs/1104.3543