Distortion problems, from Banach space geometry, ask about the possibility of distorting the norm of a Banach space in a significant way on all of its subspaces. Big Ramsey degree problems, from combinatorics, are about proving weak analogues of the infinite Ramsey theorem in sets carrying structure. Both topics come back to the seventies and are still not well understood. While their motivations are quite disjoint, both problems share a surprisingly similar flavour.

In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step forward towards the unification of those two topics, we developped an analogue of big Ramsey degrees adapted to the study of metric structures (metric spaces, Banach spaces…). Those metric big Ramsey degrees are compacts metric spaces which are invariants associated to certain monoid actions by isometry, quantifying their default of Ramseyness. We were able to prove the existence of big Ramsey degrees for certain classical metric structures and in some cases, to give an explicit description of them ; it also seems that some classical invariants from topological dynamics can be represented as big Ramsey degrees.

In this talk, I will present this theory, illustrate it on concrete examples (the Urysohn sphere and the Banach space ${\ell }_{\mathrm{\infty }}$) and give an overview of its motivations and potential applications (to Banach space theory, Ramsey theory and dynamics).