Finite combinations of Baire numbers (Q1261895)

\(Fn_ \kappa(\theta,2)\) is the collection of all partial functions \(p:\theta\to 2\) such that \(| p|<\kappa\), partially ordered by reverse inclusion. The Baire number of a partial order is the minimal cardinality of a family of dense sets that has no filter. For \(\kappa\) a regular cardinal, let \(\ell\) be the number of different values achieved by the Baire numbers of \(Fn_ \kappa(\theta,2)\) as \(\theta\) varies over all cardinals greater than or equal to \(\kappa\). In an earlier paper [J. Symb. Logic 57, No. 3, 1086-1107 (1992)], the author showed that \(\ell\) is finite, and models of set theory in which \(\ell=1\) or \(\ell=2\) are known. In the paper under review, the author uses product forcing to show that for regular \(\kappa>\omega\), for each \(n\) with \(1\leq n<\omega\) there is a model of set theory in which \(\ell= n\).