Covering a bounded set of functions by an increasing chain of slaloms

A slalom is a sequence of finite sets of length omega. Slaloms are ordered by coordinatewise inclusion with finitely many exceptions. Improving earlier results of Mildenberger, Shelah and Tsaban, we prove consistency results concerning existence and non-existence of an increasing sequence of a certain type of slaloms which covers a bounded set of functions in the Baire space