We cannot end before at least briefly discussing one other spectacular result of the 90's.

We recall that the homogeneous Banach space problem (P5) is: If X is isomorphic to all Y ⊆ X, is X isomorphic to l2? This was solved by combining two beautiful pieces of work, Gowers' dichotomy theorem (Theorem 3.1) and the following theorem of Komorowski and Tomczak-Jaegermann [KT1, KT2]. A nice exposition somewhat simplifying the argument appears in [TJ1].

8.1 Theorem. If X is homogeneous and not isomorphic to l2 then X has a subspace without an unconditional basis.

It then follows that no subspace of X has an unconditional basis and so by the dichotomy theorem plus the fact that X is homogeneous we have that X must be H.I. But in view of the result of [GM1] that an H.I. space is not isomorphic to any proper subspace, this is impossible. Thus the solution of the homogeneous Banach space problem is achieved.

Komorowski and Tomczak-Jaegermann actually prove something stronger. They show

Theorem. Let X be a Banach space not containing a subspace isomorphic to l2. Then X contains a subspace without an unconditional basis.

Even more recently Komorowski and Tomczak-Jaegermann have made substantial progress on (Q4): if all subspaces of X have an unconditional basis is X isomorphic to l2? They proved [KT3].

Theorem. If every subspace of(X ⊕ X ⊕ …)l2 has an unconditional basis then X is isomorphic to l2.

The proof of Theorem 8.2 is well exposed in [TJ1] and we shall not repeat it here. It is interesting to note that spreading models enter at one part of the argument.