Geometric duality, perfect graphs, and the Sierpiński space

Abstract

In their classical paper \emph{On the stopping time Banach space}, Bang and Odell, among a plethora of results concerning the dyadic stopping time space and its dual, presented the first non-trivial example of the \emph{duality phenomenon} between combinatorial Banach spaces. We give a full characterization of such pairs $(\mc{F}_0, \mc{F}_1)$ of families of finite sets: This duality holds iff there is a perfect graph $G$ on $\NN$ such that $\mc{F}_0$ consists of all finite cliques of $G$ and $\mc{F}_1$ consists of all finite anti-cliques of $G$. As it turns out, Lovász' famous perfect graph theorem is an immediate corollary of this result. Among the many examples of such pairs of families, we investigate a particularly interesting one, when $G$ is the Sierpiński graph, and study general methods of embedding combinatorial and classical sequence spaces in the generated space, including the Schreier and ${\ell }_p$ spaces.