  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 $\mathrm{FIN}_k$ 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 $\ell_\infty$ and
the Urysohn sphere, with an explicit characterization in the case of
$\ell_\infty$.
