Stationary sets added when forcing squares

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.