Ramsey Methods in Analysis
This book consists of two parts prepared by the authors for the Advanced Course on Ramsey Methods in Analysis held at the Centre de Recerca Matemàtica in January 2004. It is aimed at graduate students and researchers interested in the topic, which has developed significantly over the last fifteen years. Big progress was achieved in the 1990's, when several long-standing problems in Banach space theory were solved. In 1991, T. Schlumprecht published his solution of the distortion problem and T. Gowers with W. T. Maurey published their solution of the unconditional basic sequence problem. Their constructions gave rise to more general methods, which turned out to be fruitful in further research and which brought about a number of other deep and interesting results.
The first part of the book (written by S. Argyros) is an exposition of methods of construction of peculiar Banach spaces developed by a unified attitude to the particular examples of Banach spaces given by B. S. Tsirelson and T. Schlumprecht. Applications include constructions of strongly singular extensions, quasi-reflexive hereditarily indecomposable spaces, nonseparable hereditarily indecomposable spaces, as well as the study of operators on such spaces. The second part (by S. Todorcevic) is devoted to various forms of Ramsey theory. The application of finite-dimensional theory to finite representability of the unit basis of ℓp and co spaces in a basic sequence of a Banach space is explained. Infinite-dimensional Ramsey theory of finite and infinite sequences of Nash-Williams is used to get results on summability in Banach spaces and topological Abelian groups. The Ramsey theory of finite and infinite block sequences in Banach spaces by T. Gowers concludes the exposition.