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 $X_\mathcal {F}$ of $c_{00}$ with respect to the norm $\sup \{\sum _{k\in F}|x(k)|:F\in \mathcal F\}$ where $\mathcal F$ is an arbitrary (not necessarily compact) family of finite sets covering $\mathbb {N}$.

Among other results, we discuss the following:

(1) Structure theorems bonding the combinatorics of $\mathcal F$ and the geometry of $X_\mathcal {F}$ including possible characterizations and variants of the Schur property, $\ell _1$-saturation, and the lack of copies of $c_0$ in $X_\mathcal {F}$.

(2) A plethora of examples including a relatively simple $\ell _1$-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pełczyński’s universal space.

(3) The complexity of the family $\{H\subseteq \mathbb N:X_{\mathcal {F}\upharpoonright H}$ does not contain $c_0\}$.