The ZOO of combinatorial Banach spaces

We study Banach spaces induced by families of finite sets in the most natural (Schreier-like) way, that is, we consider the completion XF of c₀₀ with respect to the norm \∑k∈ F|x(k)|:F∈F\ where F is an arbitrary (not necessarily compact) family of finite sets covering N. Among other results, we discuss the following: (1) Structure theorems bonding the combinatorics of F and the geometry of XF including possible characterizations and variants of the Schur property, ₁-saturation, and the lack of copies of c₀ in XF. (2) A plethora of examples including a relatively simple ₁-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pe czy\'nski's universal space. (3) The complexity of the family ⊆:X_F H does not contain c₀\.