Cohen reals from small forcings (Q2732281)

The author improves the results of \textit{A. Rosłanowski} and \textit{S. Shelah} [``Simple forcing notions and forcing axioms'', J. Symb. Log. 62, No. 4, 1297-1314 (1997; Zbl 0952.03061)] and \textit{J. Zapletal} [``Small forcings and Cohen reals'', J. Symb. Log. 62, No. 1, 280-284 (1997; Zbl 0874.03063)]. For these improvements he introduces the following cardinal characteristic (related to the reaping number \(r\)): \(r^*={}\)the least number of perfect subsets of \(2^\omega\) such that each \(G_\delta\) subset of \(2^\omega\) either is disjoint from or contains at least one of them. He proves that \(\min\{r,r^*\}\) is above the additivity of the meager sets and (1) the least size of a poset that adds a real but fails to add an unbounded real is \(\geq r^*\), (2) the least size of a poset that adds an unbounded real but fails to add a Cohen real is \(\geq r\), and (3) the least size of a poset that adds a real but fails to add a Cohen real is above the additivity of the meager sets.