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 ℬ or the Ramsey property ℛ. In this paper it is shown that a finite support iteration of length ω 1 of Cohen forcing, starting from L , yields a model for CH + $ \sum ^1 _2 $ () + $ \neg \sum ^1 _2 $ (ℬ) + $ \neg \sum ^1 _2 $ (ℛ).