Stationary sets added when forcing squares

Abstract

Current research in set theory raises the possibility that $\square _{\kappa ,<\lambda }$ can be made compatible with some stationary reflection, depending on the parameter $\lambda$. The purpose of this paper is to demonstrate the difficulty in such results. We prove that the poset ${\mathbb {S}}(\kappa ,<\lambda )$, which adds a $\square _{\kappa ,<\lambda }$-sequence by initial segments, will also add non-reflecting stationary sets concentrating in any given cofinality below $\kappa$. We also investigate the CMB poset, which adds $\square _\kappa ^*$ in a slightly different way. We prove that the CMB poset also adds non-reflecting stationary sets, but not necessarily concentrating in any cofinality.