2025/01/30 by Tristan Bice, Bice, Tristan, Noé de Rancourt +5 · 1 voice · 1 citation
#math.MG #math.CO #math.LO
paper · pdf · doi:10.48550/arxiv.2501.18552
We prove oscillation stability for the Banach space ℓ_∞: 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 ℓ_∞. 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.