On Ramsey-type properties of the distance in nonseparable spheres
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914906655162368 |
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| author | Koszmider, Piotr |
| author_facet | Koszmider, Piotr |
| contents | Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an uncountable subset $\mathcal Y$ of a nonseparable Banach space does not admit an uncountable $\mathcal Z\subseteq \mathcal Y$, where any two points are distant by more than $r>0$, is it because $\mathcal Y$ is the countable union of sets of diameters not bigger than $r$?
We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to $2^ω$ related to uncountable $(1+)$-separated sets, equilateral sets or Auerbach systems.
The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA+MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of Hájek, Kania, Russo for densities above $2^ω$ as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to $2^ω$ obtained previously by the author and Guzmán, Hrušák, Ryduchowski and Wark.
It remains open if the dichotomies may consistently hold for all Banach spaces of the first uncountable density or if the strong anti-Ramsey properties of the distance on the unit sphere of a Banach space can be obtained in ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07668 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Ramsey-type properties of the distance in nonseparable spheres Koszmider, Piotr Logic Functional Analysis General Topology Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an uncountable subset $\mathcal Y$ of a nonseparable Banach space does not admit an uncountable $\mathcal Z\subseteq \mathcal Y$, where any two points are distant by more than $r>0$, is it because $\mathcal Y$ is the countable union of sets of diameters not bigger than $r$? We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to $2^ω$ related to uncountable $(1+)$-separated sets, equilateral sets or Auerbach systems. The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA+MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of Hájek, Kania, Russo for densities above $2^ω$ as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to $2^ω$ obtained previously by the author and Guzmán, Hrušák, Ryduchowski and Wark. It remains open if the dichotomies may consistently hold for all Banach spaces of the first uncountable density or if the strong anti-Ramsey properties of the distance on the unit sphere of a Banach space can be obtained in ZFC. |
| title | On Ramsey-type properties of the distance in nonseparable spheres |
| topic | Logic Functional Analysis General Topology |
| url | https://arxiv.org/abs/2308.07668 |