Isomorphic embeddings of Banach spaces and universality questions

Universality questions abound in mathematics. In the most general form they are formulated as follows: given a class C of objects and a notion of quasi-order <= between them, find a subclass D of smallest cardinality which has the property that every element of C is <= an element of D. An example of this in the theory of Banach spaces is the search for an isomorphically or an isometrically universal element in the class of Banach spaces in a given density. There are interesting independence results in the theory of isomorphically universal Banach spaces (see the work of Brech and Koszmider) and on the other hand, model theoretic results (Shelah and Usvyatsob) in the theory of isometrically universal spaces. The latter have the advantage that in a certain sense they are absolute, specifically, they depend only on cardinal arithmetic and not on the set-theoretic universe, but a disadvantage that they cannot talk about isomorphism, only about isometry.

We would like to have means to obtain similar "semi-absolute" results in the isomorphic theory, and we present some initial results in this direction.