Abstract. We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ∞/c 0 is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.We also prove that it is consistent that there are universal Banach spaces of density c, but ℓ∞/c 0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into ℓ∞/c 0 .

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/f in has under CH and in the ℵ 2 -Cohen model. We prove a similar result in the category of Banach spaces.

Abstract. We strengthen the property Δ of a function f : [ω 2 ] 2 → [ω 2 ] ≤ω considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces K as above where K n is hereditarily separable for each n ∈ N. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space C(K) is an Asplund space of density ℵ 2 which has no Fréchet smooth renorming, nor an uncountable biorthogonal system.

We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.