Oscillation stability by the Carlson-Simpson theorem

TL;DR

This work establishes oscillation stability results for two fundamental metric structures via Carlson–Simpson dual Ramsey theory. It proves that every weak-* Borel, uniformly continuous map on the unit sphere of $oldsymbol{\,l_oldsymbol{ackslashfty}}$ stabilizes on a suitable isometric subcopy, and it provides a shorter combinatorial proof of stability for the Urysohn sphere by leveraging a universal pseudometric space and the same Ramsey machinery. The results connect oscillation stability with metric big Ramsey degrees and illustrate the power of finite-density, dual Ramsey arguments in continuous structures. The definability assumption (weak-* Borel) is shown to be essential in at least one case, and the techniques open a path toward metric Ramsey theory applications to broader classes of continuous spaces.

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 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.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2: Nguyen Van Thé--Sauer 2009
• Definition 2.1
• Theorem 2.2: Carlson--Simpson 1984
• Proposition 3.1
• proof : Proof of Theorem \ref{['thm:OscStabEllInf']}
• Lemma 3.2
• proof
• proof : Proof of Proposition \ref{['prop:MainLInf']}
• Proposition 3.3