Making doughnuts of Cohen reals

Abstract

AbstractFor 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 𝒟 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 ℬ︁ or the Ramsey property ℛ. 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 + $ \sum ^1 _2 $(𝒟) + $ \neg \sum ^1 _2 $(ℬ︁) + $ \neg \sum ^1 _2 $(ℛ).