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 $\ell_\infty$.

翻译：振荡稳定性是巴拿赫空间理论中的一个重要概念，与离散拉姆齐理论密切相关。例如，高尔斯利用他著名的$\mathrm{FIN}_k$拉姆齐定理作为关键工具，证明了巴拿赫空间$c_0$的振荡稳定性。我们发展了这种关联背后的理论，并引入了紧致大拉姆齐度的概念，将（离散）大拉姆齐度的理论进行了推广。随后，我们证明了巴拿赫空间$\ell_\infty$和乌雷松球面存在紧致大拉姆齐度，并在$\ell_\infty$的情形下给出了显式刻画。