Distinctness of Tukey classes for sequential structures at countable cofinality

Determine whether, for every infinite cardinal κ = aleph_α with cofinality ω, all Tukey classes described for the sequential structures of separable metrizable spaces of size κ are pairwise distinct under the Tukey order. Specifically, ascertain whether the relations P(κ) × (λ, =) + Q(κ) × (μ, =) + S(κ) (for all (λ, μ) in Δ = { (λ, μ) : 0 ≤ λ < ω and μ ≤ ω }), together with P(κ) × (aleph_β, =) + S(κ) for each β < α and P(κ) × (κ, =), are mutually non-equivalent under Tukey order. Here P(κ) = ((([κ]^{<ω})^ω)_∞, i_∞) with i_∞ meaning coordinatewise intersection at infinitely many indices; Q(κ) = 0 if cof(κ) ≠ ω, and otherwise Q(κ) = ((∏_n [κ_n]^{<ω})_∞, i_∞) for any strictly increasing sequence (κ_n) converging to κ; and S(κ) = 0 if cof(κ) ≠ ω, and otherwise S(κ) = ∑_n P(κ_n) × (κ_n, =).

Background

The paper classifies Tukey types of relations that encode sequential structures of separable metrizable spaces. For spaces of size κ = aleph_α with cof(κ) ≠ ω, there is a unique Tukey class P(κ) × (κ, =). For κ = aleph_α with cof(κ) = ω, the authors present a detailed family of Tukey classes parameterized by Δ (a finite/countable parameter controlling small neighborhood sizes) together with an ordinal tail indexed by β ≤ α.

These classes are expressed using relations P(κ), Q(κ), and S(κ), which are built from products and sums of sequence-of-finite-subset structures under the relation i_∞ (infinitely many coordinate intersections). Although the structure and bounds are established, it remains unresolved whether all these parameterized classes are actually distinct, i.e., whether any two yield non-bimorphic Tukey types.