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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.18552 |
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| _version_ | 1866909470199644160 |
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| author | Bice, Tristan de Rancourt, Noé Hubička, Jan Konečný, Matěj |
| author_facet | Bice, Tristan de Rancourt, Noé Hubička, Jan Konečný, Matěj |
| contents | We prove oscillation stability for the Banach space $\ell_\infty$: every weak-* Borel, uniformily continuous map from the unit sphere of this space to a compact metric space can be made arbitrarily close to a constant map when restricted to the unit sphere of a suitable linear isometric subcopy of $\ell_\infty$. We also give a new proof of oscillation stability for the Urysohn sphere (a result by Nguyen Van Thé--Sauer): every uniformily continuous map from the Urysohn sphere to a compact metric space can be made arbitrarily close to a constant map when restricted to a suitable isometric subcopy of the Urysohn sphere. Both proofs are based on Carlson-Simpson's dual Ramsey theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oscillation stability by the Carlson-Simpson theorem Bice, Tristan de Rancourt, Noé Hubička, Jan Konečný, Matěj Metric Geometry Combinatorics Logic 51F30, 03E02, 05C55, 05D10, 46B99, 46T99 We prove oscillation stability for the Banach space $\ell_\infty$: every weak-* Borel, uniformily continuous map from the unit sphere of this space to a compact metric space can be made arbitrarily close to a constant map when restricted to the unit sphere of a suitable linear isometric subcopy of $\ell_\infty$. We also give a new proof of oscillation stability for the Urysohn sphere (a result by Nguyen Van Thé--Sauer): every uniformily continuous map from the Urysohn sphere to a compact metric space can be made arbitrarily close to a constant map when restricted to a suitable isometric subcopy of the Urysohn sphere. Both proofs are based on Carlson-Simpson's dual Ramsey theorem. |
| title | Oscillation stability by the Carlson-Simpson theorem |
| topic | Metric Geometry Combinatorics Logic 51F30, 03E02, 05C55, 05D10, 46B99, 46T99 |
| url | https://arxiv.org/abs/2501.18552 |