Making doughnuts of Cohen reals

For a ? b ? ω with b\ a infinite, the set D = {x ∈ ω]^ω : a ? x ? b} is called a doughnut. A set S ? ω]^ω has the doughnut property ?? if it contains or is disjoint from a doughnut. It is known that not every set S ? ω]^ω has the doughnut property, but S has the doughnut property if it has the Baire property ?? or the Ramsey property ?. In this paper it is shown that a finite support iteration of length ω₁ of Cohen forcing, starting from L , yields a model for CH + (??) + (??) + (?).