Kubis and Szeptycki introduced the notion of n-Ramsey spaces for every n\in\omega which is a natural n-dimensional generalization of sequential compactness. They also showed that under CH there are examples of n-Ramsey spaces that are not (n+1)-Ramsey.

We will discuss a joint work with Osvaldo Guzman and Carlos Lopez Callejas in which we improved the previous result by weakening the assumption of CH. We will also deal with other related results involving cardinal invariants that become relevant in this context.

Then we will introduce a further generalization of n-Ramsey spaces to barriers of arbitrary rank and discuss which of the previous results remain true in the infinite case. (The last part is ongoing work, also joint with Szeptycki and Memarpanahi).