One of the acknowledged major successes of set theory in analysis has been the use of Ramsey theory in the study of Banach spaces. The first such use was the concept of a spreading model of a Banach space due to Brunel and Sucheston,. This was a way of joining the finite and infinite dimensional structure of a Banach space in an asymptotic manner. Perhaps the best known of these applications of Ramsey theory is Rosenthal's theorem: A Banach space $Y$ does not contain an isomorphic copy of $l_1$ if and only if every bounded sequence in $Y$ has a weakly Cauchy subsequence. Closely related to this is a subsequent theorem of Rosenthal about pointwise compact sets of
functions: Any sequence of continuous functions on a Polish space which is pointwise bounded, and every cluster point of which is a Borel function, has a converging subsequence. Shortly afterwards Bourgain, Fremlin,and Talagrand have extended this result to the conclusion that the sequence is actually sequentially dense in its pointwise closure. This, along with earlier work of Odell and Rosenthal, resulted in a renewed interest in spaces of Baire-class 1 functions; namely, pointwise limits of continuus functions. Using the results of Bourgain, Fremlin and Talagrand, Godefroy showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space $K$ is representable as a compact set of Baire class-1 functions then so is $P(K)$, the space of all Radon probability measures on $K$ with the weak$^*$ topology. Recent results have been obtained towards a fine structure theory of compact sets of first Baire class by Todorcevic; in particular, every compact set of first Baire class contains a dense metrizable subspace.

The involvement of infinite dimensional Ramsey theory was recently lifted to a higher level of sophistication by W.~T.~Gowers in his positive solution to the homogeneous space problem of Banach: if a Banach space is isomorphic to all of its infinite dimensional subspaces then it is isomorphic to a Hilbert space. The Ramsey theoretic part of his result, stating that every Banach space contains a subspace which either has an unconditional basis or is hereditarily indecomposable was combined with some analytical work of Komorowski and Tomczak-Jaegermann.

The strongest potential Ramsey theorem in a Banach space setting can be stated as: is every uniformly continuous real valued function on a unit sphere $S_X$ of a Banach space $X$ oscillation stable? This is called the distortion problem. It was solved by Odell and Schlumprecht in the 90's in the negative. However the existence of a distortable space of "bounded distortion" remains unkown. Such a space could lead to a new form of a weak Ramsey theorem of some sort. Interesting work on this problem has been done by Odell, Schlumprecht, Maurey, V. Milman, Tomczak-Jaegermann, among others.

Recent solutions of the two most famous problems in this area of Banach space theory, the distortion problem (Odell and Schlumprecht) and the unconditional basic sequence problem (Gowers and Maurey), are closely tied to a deeper understanding of a particular example of a non-classical Banach space due to Tsirelson. The inductive definition of its norm involving ``admissible families'' of sets makes the space susceptible to a set theoretical analysis where the notion of ``admissible'' appears with a different name ``relatively small'' (Ketonen-Solovay). Gowers' Ramsey-theoretic dichotomy for Banach spaces also seems susceptible to a further set-theoretical analysis, in particular in the direction of Ellentuck-type theorems, which are so abundant in the infinite dimensional Ramsey theory. Bringing together people in these two areas will very likely result in a much better understanding of these

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