Big Ramsey degrees in the metric setting
share
Oscillation stability is an important concept in Banach space theory which happens to be closely connected to discrete Ramsey theory. For example, Gowers proved oscillation stability for the Banach space c_0 using his now famous Ramsey theorem for FIN_k as the key ingredient. We develop the theory behind this connection and introduce the notion of compact big Ramsey degrees, extending the theory of (discrete) big Ramsey degrees. We then prove existence of compact big Ramsey degrees for the Banach space ℓ_∞ and the Urysohn sphere, with an explicit characterization in the case of ℓ_∞.
READ FULL TEXT