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 &Dscr; 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 &Bscr; or the Ramsey property ℛ. In this paper it is shown that a finite support iteration of length ω1 of Cohen forcing, starting from <BI>L</BI>, yields a model for CH + <UEQN>$ \sum ^1 _2 $</UEQN>(&Dscr;) + <UEQN>$ \neg \sum ^1 _2 $</UEQN>(&Bscr;) + <UEQN>$ \neg \sum ^1 _2 $</UEQN>(ℛ). [ABSTRACT FROM AUTHOR]