Geometric duality, perfect graphs, and the Sierpiński space
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866910218603986944 |
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| author | Borodulin-Nadzieja, Piotr Farkas, Barnabás Pelczar-Barwacz, Anna |
| author_facet | Borodulin-Nadzieja, Piotr Farkas, Barnabás Pelczar-Barwacz, Anna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric duality, perfect graphs, and the Sierpiński space Borodulin-Nadzieja, Piotr Farkas, Barnabás Pelczar-Barwacz, Anna Functional Analysis Combinatorics Logic 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. |
| title | Geometric duality, perfect graphs, and the Sierpiński space |
| topic | Functional Analysis Combinatorics Logic |
| url | https://arxiv.org/abs/2605.14072 |