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Monday, December 13, 2004

**Abstract:** Banach space theory underwent an explosive development a decade ago with discoveries of new exotic spaces using methods of W.T. Gowers and B. Maurey and solutions to many of the classical problems by several people. However, this also lead to new problems and hope for a better understanding of the isomorphism classes of subspaces of any given Banach space. We prove a dichotomy for Banach spaces saying that any infinite-dimensional Banach space contains either a minimal subspace or a continuum of incomparable subspaces. The proof relies heavily on the Ramsey type methods invented by W.T. Gowers and the solution of the distortion problem by E. odell and T. Schlumprecht, but also introduces new methods from logic.