More on setwise climbability properties

We introduce two types of variations of setwise climbability properties, which have been introduced by the second author as fragments of Jensen's square principles. We show that variations of the first type are equivalent to known principles and that they are consistent with the Proper Forcing Axiom(PFA). On the other hand, those of the second type can be characterized as Martin-type axioms for some classes of posets defined in terms of a new variation of generalized Banach-Mazur games, and they are no longer consistent with PFA. We also study how large fragments of PFA are consistent with these principles.