Ramsey Theory And Banach Space Geometry

Abstract

We present two main results, one related to the original construction of Tsirelson's space and one that elaborates on a Ramsey type theorem formulated and proved by Gowers to obtain the stabilization of Lipschitz functions on the unit sphere of c0 . We obtain an expression for the norm of the space constructed by Tsirelson. This study of the original construction is aimed at providing a description of the space that could lend itself to the use of tools coming from Ramsey theory. The expression can be modified to give the norm of the dual of any mixed Tsirelson space. In particular, our results can be adapted to give the norm for the dual of the space S constructed by Schlumprecht. We give a constructive proof of the finite version of Gowers' FINk Theorem and analyse the corresponding upper bounds. We compare the finite FINk Theorem with the finite stabilization principle in the case of spaces of the form n [INFINITY], n ∈ N and establish a more slowly growing upper bound for the finite stabilization principle in this particular case.