This is an announcement for the paper "An infinite Ramsey theorem and some Banach-space dichotomies" by W. T. Gowers. Abstract: A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and Tomczak-Jaegermann, this gives a positive answer to Banach's problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20 (Primary) 03E02, 03E15, 05D10, 46B03 (Secondary) Citation: Ann. of Math. (2), Vol. 156 (2002), no. 3, 797--833 Remarks: 37 pages, published version The source file(s), ArxivGowers.tex: 109202 bytes, amltd2004.sty: 33983 bytes, is(are) stored in gzipped form as 0501105.tar.gz with size 42kb. The corresponding postcript file has gzipped size 112kb. Submitted from: wtg10 at dpmms.cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0501105 or http://arXiv.org/abs/math.FA/0501105 or by email in unzipped form by transmitting an empty message with subject line uget 0501105 or in gzipped form by using subject line get 0501105 to: math at arXiv.org.