Some convergence and compactness properties with a Ramsey theoretic flavour

Ramsey's and Nash-William's Theorems suggest higher dimensional variations of the classical notion of a (one-dimensional) convergent sequence. I'll survey some of the ideas and results that have been obtained in the last couple of years with various co-authors. Some of these notions have already been discussed at this seminar (Cesar Corral's talks at the Pitt Topology Seminar in 2024) so I will focus on results obtained since then.