A Remark on Schimmerling’s Question

Abstract

Schimmerling asked whether \(\square ^{\ast }_{\lambda }+\mathsf {GCH}\) entails the existence of a λ+-Souslin tree, for a singular cardinal λ. We provide an affirmative answer under the additional assumption that there exists a non-reflecting stationary subset of \({E}^{{\lambda }^{+}}_{{\neq } \text {cf}(\lambda )}\). As a bonus, the outcome λ+-Souslin tree is moreover free.