J. L. Krivine's theorem states that for every Banach space X with a basis, there exists a p in [1, infinity] such that l_p is finitely block represented in X. The set of all such p's is called the Krivine set of X. As it was proved by H.P. Rosenthal, this set is stabilized on some block subspace Y of X, i.e. the Krivine set of Y and the corresponding one of any of its further block subspaces coincide. The form of such a stabilized Krivine set has been a subject of study, since Rosenthal asked whether it always had to be a singleton. This question was answered negatively by E. Odell and Th. Schlumprecht by constructing a space having [1, infinity] as its stabilized Krivine set.

The question that followed was if such a stabilized Krivine set had to be an interval, which was asked by P. Habala and N. Tomczak-Jaegermann as well as by E. Odell. We answer this question in the negative direction by constructing, for every subset F of [1,infinity] which is either finite or consists of an increasing sequence and its limit, a reflexive Banach space X with an unconditional basis such that for every infinite dimensional block subspace Y of X, the Krivine set of Y is precisely F.

This construction also addresses some open problems concerning spreading models.

This is joint work with K. Beanland and D. Freeman. Speaker(s): Pavlos Motakis (Texas A&M University)