Forbidden rectangles in compacta

We establish negative results about “rectangular” local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type ω × ω 2 . For another, the compactum βω has no nontrivially rectangular local bases, and the same is consistently true of β ω ∖ ω : no local base in βω has cofinal type κ × c if κ < m σ - n - linked for some n ∈ [ 1 , ω ) . Also, CH implies that every local base in β ω ∖ ω has the same cofinal type as one in βω. o answer a question of Dobrinen and Todorčević about cofinal types of ultrafilters: the Fubini square of a filter on ω always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on ω is commutative modulo cofinal equivalence.