In many countable discrete structures, the obvious analogue of the infinite Ramsey theorem doesn't hold, but one can still prove a weak version of it where every colouring reduces to a finite number of colors on a subcopy. This number of colors is called a big Ramsey degree; big Ramsey degrees have been computed for many structures from the seventies to the recent years.

Motivated by several results and problems from topology, functional analysis, and dynamics, in a joint work with Tristan Bice, Jan Hubička and Matěj Konečný, we designed a notion of big Ramsey degrees adapted to the study of separable metric structures. In this setting, big Ramsey degrees are not anymore numerical invariants but compact metric spaces. We characterized the big Ramsey degrees of the Urysohn sphere and the Banach space $\ell_\infty$; interesting dynamical objects can also be represented as big Ramsey degrees. In this talk, I will present the theory, some of its motivations, and our results.