Noé de Rancourt , Université de Lille
Big Ramsey degrees of metric structures
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 structures such as hypergraphs, partially ordered sets, etc. Those two topics, coming back to the seventies, have quite disjoint motivations but share a surprisingly similar flavour. In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step 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...). Our theory allows us to associate to certain metric structures a sequence of compact metric spaces quantifying their default of Ramseyness. In this talk, I'll present our theory and its motivations and illustrate it on the examples of the Banach space $\ell_\infty$ and the Urysohn sphere. If time permits, links with topological dynamics will also be discussed.
QNC 1507 or Join on Zoom