Oscillation stability by the Carlson-Simpson theorem

TL;DR

This paper establishes oscillation stability results for the Banach space lex_ and the Urysohn sphere, using Carlson-Simpson's dual Ramsey theorem to show maps can be approximated by constants on suitable subspaces.

Contribution

It provides new proofs of oscillation stability for lex_ and the Urysohn sphere, extending the application of Carlson-Simpson's dual Ramsey theorem.

Findings

Oscillation stability for lex_ proven using dual Ramsey theorem.

New proof of oscillation stability for the Urysohn sphere.

Both results show maps can be approximated by constants on suitable subspaces.

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.