Existence of ultrafilters that are A−-filters but not A-filters

Determine whether there exists an ultrafilter U on ω that is an A−-filter but not an A-filter; that is, establish whether there exists U such that for any family F ⊆ U with |F| ≤ ω1 there exists an infinite A ⊆ ω with A ⊆* G for all G ∈ F (A−-filter property), while U fails the A-filter property that for any family {Fα: α < ω1} ⊆ U there exist sets Gn ∈ U (n ∈ ω) such that for each α there is n with Gn ⊆ Fα.

Background

The paper introduces strengthened and weakened variants of the A-filter notion: an A+-filter requires, for families of size ≤ ω1, the existence of an element of the filter that is ⊆-below all family members, while an A−-filter requires the existence of an infinite subset of ω with this ⊆ property. Every A+-filter is an A-filter, and every A-filter is an A−-filter, but the strict separations between these classes—especially at the level of ultrafilters—are not fully understood.

The authors provide examples showing that certain non-ultrafilters can be A− but not A, and relate these notions to the cardinal p. Whether there exists an ultrafilter with the A− but not A property remains unresolved and is important for understanding the interaction between combinatorial filter properties and the topological constructions central to Rω1-factorizability.