Making doughnuts of Cohen reals

For subsets of natural numbers a and b where b-a is infinite, the set of all infinite sets which are contained in b and containing a is called a doughnut. A set S of infinite subsets of natural numbers has the doughnut property, if it contains or is disjoint from a doughnut. It is known that not every such set S 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 omega₁ of Cohen forcing, starting from L, yields a model for CH + "all Sigma-1-2 sets have the doughnut property" + "not all Sigma-1-2 sets are Ramsey" + "not all Sigma-1-2 sets have the Baire property".

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