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 $X_{\text{mc}}F$ of $c_{00}$ with respect to the norm $\sup \{{\sum }_{k\in F}\mathrm{\mid }x(k)\mathrm{\mid }:F\in \text{mc}F\}$ where $\text{mc}F$ is an arbitrary (not necessarily compact) family of finite sets covering $\text{mbb}N$. Among other results, we discuss the following: (1) Structure theorems bonding the combinatorics of $\text{mc}F$ and the geometry of $X_{\text{mc}}F$ including possible characterizations and variants of the Schur property, ${\mathrm{\ell }}_1$-saturation, and the lack of copies of $c_0$ in $X_{\text{mc}}F$. (2) A plethora of examples including a relatively simple ${\mathrm{\ell }}_1$-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 $\{H\subseteq \text{NN}:X_{\text{mc}F\upharpoonright H}$ does not contain $c_0\}$.