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Simultaneous nonvanishing for all k ≥ 2 with dominating number d = aleph_{ω+1}

Ascertain whether there exists a model of ZFC in which the dominating number \mathfrak{d} equals \aleph_{\omega+1} and, simultaneously, \lim^k A \neq 0 holds for every integer k with 2 \leq k < \omega, where A is the inverse system of abelian groups defined from the sets I_f = {(k,m) \in \omega \times \omega \mid m < f(k)} with groups A_f = \bigoplus_{I_f} \mathbb{Z} and restriction bonding maps.

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Background

Using square principles and weak diamonds, the paper constructs models where \limk A \neq 0 holds simultaneously for all 2 \leq k < \omega, achieving \mathfrak{d} = \aleph_{\omega+2}. By Goblot’s theorem, such nonvanishing requires \mathfrak{d} \geq \aleph_{\omega+1}.

The authors state that matching this with the optimal lower bound \mathfrak{d} = \aleph_{\omega+1} is still unresolved.

References

"By the aforementioned result of Goblot, $\bigwedge_{2 \leq k < \omega} \limk {A} \neq 0$ implies that $\mathfrak{d}$ is at least $\aleph_{\omega+1}$. Our result is thus almost optimal; it remains open whether one can obtain the same conclusion with $\mathfrak{d} = \aleph_{\omega+1}$."

— Simultaneously nonvanishing higher derived limits (2411.15856 - Casarosa et al., 24 Nov 2024) in Introduction, discussion following Theorem B