Forcing and construction schemes (Q2330055)

Let \(P_\omega (\omega_1)\) denote the set of all finite subsets of \(\omega_1\), ordered by inclusion. A \textit{construction scheme} is, roughly speaking, a highly structured infinite partition \(\mathcal{F}\) of a cofinal subset of \(P_\omega (\omega_1)\). For an integer \(n \geq 2\), \(\mathcal{F}\) is \(n\)-\textit{capturing} if for any uncountable \(\Delta\)-system \(S\) of finite subsets of \(\omega_1\), \(\mathcal{F}\) captures \(n\) of its members. The point is that capturing construction schemes can be used as an alternative to forcing to construct objects such as Banach spaces, Suslin trees, gaps, etc. The authors show that (a) by adding uncountably many Cohen reals, one adds a construction scheme that is \(n\)-capturing for all \(n\), and (b) the existence of an \(n\)-capturing construction scheme is consistent with the nonexistence of any \((n + 1)\)-capturing construction scheme. They also compare the \(n\)-capturing hierarchy with the \(m\)-Knaster hierarchy.