Open problems in Banach spaces, universality

Question

I have gathered a list of universality problems in Banach spaces which have been solved:

1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces.

2.If a space is universal for the class of separable reflexive spaces, then it is universal for the class of separable Banach spaces.

3.There is a separable reflexive space which is universal for all separable uniformly convex spaces.

My question is, are there any open problems in a similar vein to these?

Answer 1

It is known that the collection of all separable Banach spaces not containing $\ell_1$ does not have a universal member, but it is not known if this collection strongly bounded. That is, if you consider an analytic (in SB) subset of this collection is there an space not containing $\ell_1$ into which every member of this subset embedds. The obstruction for this problem is to show that every separable space not containing $\ell_1$ embedds into a space with a basis also not containing $\ell_1$. The Zippin type embedding theorems only hold for reflexive spaces and spaces with separable duals.

On a related note, it is not known if every separable uniformly convex space embedds into a uniformly convex space with a basis.