Metacompact subspaces of products of ordinals

Let $X$ be a subspace of the product of finitely many ordinals. $X$ is countably metacompact, and $X$ is metacompact iff $X$ has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: $X$ is $\lambda$-metacompact iff $X$has no closed subset homeomorphic to a $(\kappa_1, \ldots , \kappa_n)$-stationary set where $\kappa_1 < \lambda$.