Point-cofinite covers in the laver model

Let S₁(Γ, Γ) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. b is the minimal cardinality of a set of reals not satisfying S₁(Γ, Γ). We prove the following assertions: (1) If there is an unbounded tower, then there are sets of reals of cardinality b satisfying S₁(Γ, Γ). (2) It is consistent that all sets of reals satisfying S₁(Γ, Γ) have cardinality smaller than b. These results can also be formulated as dealing with Arhangel'skiǐ's property α₂ for spaces of continuous real-valued functions. The main technical result is that in Laver's model, each set of reals of cardinality b has an unbounded Borel image in the Baire space ω^ω.