Suslin forests at non-weakly compact regular cardinals

Determine whether, under V=L, a Suslin (κ, κ^+)-forest exists for every uncountable regular cardinal κ that is not weakly compact.

Background

Under V=L, every uncountable regular cardinal that is not weakly compact carries a κ-Suslin tree, while weak compactness characterizes the obstruction to such trees. A Suslin (κ, κ+)-forest naturally yields κ-Suslin trees, so the paper asks whether the same weak-compactness characterization holds for forests. The unresolved issue is whether the forest existence principle is equally broad at all non-weakly compact regular cardinals.

References

Under $\mathsf{V=L}$, is there a Suslin $(\kappa, \kappa+)$-forest for every uncountable regular cardinal $\kappa$ which is not weakly compact?

— Quagmires and large Suslin forests  (2608.28505 - Notaro, 28 Aug 2026) in Section 4, “Questions”

A separate problem is to recover a proof of Laver's result that the existence of a Suslin $(\omega_1, \omega_2)$-forest follows from Silver's principle $W_{\omega_1}(\omega_2)$ together with $\diamondsuit$. We expect a construction using only $\diamondsuit + W_{\omega_1}(\omega_2)$ not to produce the coherence obtained in Eskew's construction.

\begin{problem} Prove Laver's result that the existence of a Suslin $(\omega_1, \omega_2)$-forest follows from $\diamondsuit + W_{\omega_1}(\omega_2)$. \end{problem}

— Quagmires and large Suslin forests  (2608.28505 - Notaro, 28 Aug 2026) in Section 4, “Questions”