Selection principles and the minimal tower problem

We study diagonalizations of covers using various selection principles, where the covers are related to linear quasiorderings (τ -covers). This includes: equivalences and nonequivalences, combinatorial characterizations, critical cardinalities and constructions of special sets of reals. This study leads to a solution of a topological problem which was suggested to the author by Scheepers (and stated in [15]) and is related to the Minimal Tower problem. We also introduce a variant of the notion of τ -cover, called τ -cover, and settle some problems for this variant which are still open in the case of τ -covers. This new variant introduces new (and tighter) topological and combinatorial lower bounds on the Minimal Tower problem.