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arXiv · 2501.18552

Oscillation stability by the Carlson-Simpson theorem

Tristan Bice·Noé de Rancourt·Jan Hubička·Matěj Konečný
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Abstract

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 Th\'e--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.

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Tristan Bice, Noé de Rancourt, Jan Hubička, Matěj Konečný. 2025-01-30. Oscillation stability by the Carlson-Simpson theorem. https://arxiv.org/abs/2501.18552

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