Abstract We construct a model in which all C -sequences are trivial, yet there exists a -Souslin tree all of whose limit levels are vanishing levels. This provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen’s 1978 forcing construction, and by analyzing a gallery of -cc forcing extensions of the outcome model.
Rinot et al. (Thu,) studied this question.