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arXiv · 2605.14072

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

Piotr Borodulin-Nadzieja·Barnabás Farkas·Anna Pelczar-Barwacz
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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.

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Piotr Borodulin-Nadzieja, Barnabás Farkas, Anna Pelczar-Barwacz. 2026-05-13. Geometric duality, perfect graphs, and the Sierpi\'nski space. https://arxiv.org/abs/2605.14072

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