Canonical equivalence relations on nets of PSc0PS_{c_0}

We give a list of canonical equivalence relations on discrete nets of the positive unit sphere of $c_0$. This generalizes results of W. T. Gowers and A. D. Taylor.Comment: 41 page

A c_0c\_0-saturated Banach space with no long unconditional basic sequences

We present a Banach space $\mathfrak X$ with a Schauder basis of length $\omega\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak X$ have ``few operators'' in the sense that every bounded operator $T:X \to \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $\omega\_1$-singular operator, i.e., an operator $S$ which is not an isomorphism on any non-separable subspace of $X$. We also show that while $\mathfrak{X}$ is not distortable (being $c\_0$-saturated), it is arbitrarily $\omega\_1$-distortable in the sense that for every $\lambda>1$ there is an equivalent norm $\||\cdot \||$ on $\mathfrak{X}$ such that for every non-separable subspace $X$ of $\mathfrak{X}$ there are $x,y\in S\_X$ such that \||\cdot \|| / \||\cdot \||\ge \la

Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal

Banach spaces and Ramsey Theory: some open problems

We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.Comment: 17 pages, no figures; RACSAM, to appea

The oscillation stability problem for the Urysohn sphere: A combinatorial approach

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space \Ur. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.Comment: 19 page