High dimensional countable compactness and ultrafilters

We define several notions of a limit point on sequences with domain a barrier in $[\omega]^{<\omega}$ focusing on the two dimensional case $[\omega]^2$. By exploring some natural candidates, we show that countable compactness has a number of generalizations in terms of limits of high dimensional sequences and define a particular notion of $\alpha$-countable compactness for $\alpha\leq\omega_1$. We then focus on dimension 2 and compare 2-countable compactness with notions previously studied in the literature. We present a number of counterexamples showing that these classes are different. In particular assuming the existence of a Ramsey ultrafilter, a subspace of $\beta\omega$ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.