Embeddings into Banach spaces with finite dimensional decompositions

We consider the following general problem: Given a class $\mathcal C$ of Banach spaces, is there an element $X$ of $\mathcal C$, or in a class closely related to $\mathcal C$, which is universal for the class $\mathcal C$, meaning that every member of $\mathcal C$ is isomorphically a subspace of $X$? In many cases these type questions can be easily solved in the category of spaces having a basis, or more generally, a finite dimensional decomposition (FDD). Then, the aforementioned problem becomes a problem of the following type: Can a Banach space $X$ in certain class $\mathcal C$ be embedded into a space $Z$ of that class, or to a class closely related to $\mathcal C$, with $Z$ having a basis or an FDD? <br/> <br/>In our talk we will present a general combinatorial argument leading to the solutions of these type of problems. Secondly, we present the solution some concrete problems using our machinery: <br/> <br/>(1) Intrinsic characterization of subspaces of $\ell_p$-sums of finite dimensional subspaces (Question by W.B. Johnson in 1977). <br/>(2) Existence of a separable reflexive Banach space containing all separable super reflexive Banach spaces (Question by J.~Bourgain in 1980). <br/>(3) Existence of separable reflexive spaces being universal for the class of separable reflexive spaces with given Szlenk index (Problem by A.~Pe{\l}czy{\'n}ski 2005) <br/> <br/>The work presented is joint work with E.~Odell (for (1) and (2)), and joint work with E.~Odell and A.~Zs\'ak (for (3)).