Oscillation stability by the Carlson-Simpson theorem

We prove oscillation stability for the Banach space ${\mathrm{\ell }}_{\mathrm{\infty }}$: 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 ${\mathrm{\ell }}_{\mathrm{\infty }}$. 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.