Geometric duality, perfect graphs, and the Sierpi\'nski space

TL;DR

This paper characterizes duality pairs of finite set families via perfect graphs, linking combinatorial Banach space duality to graph theory, and explores embeddings of classical spaces using the Sierpiński graph.

Contribution

It provides a full characterization of duality pairs of finite set families through perfect graphs, connecting Banach space duality with graph theory and space embeddings.

Findings

Duality holds iff families are finite cliques and anti-cliques of a perfect graph.

Lovász' perfect graph theorem follows from this characterization.

Studied embeddings of classical spaces like Schreier and ℓ_p in the constructed spaces.

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, Lov\'asz' 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 Sierpi\'nski graph, and study general methods of embedding combinatorial and classical sequence spaces in the generated space, including the Schreier and $\ell_p$ spaces.