On a convenient property about $${[\gamma]^{\aleph_0}}$$

Abstract

Several situations are presented in which there is an ordinal γ such that \({\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}\) is a stationary subset of \({[\gamma]^{\aleph_0}}\) for all stationary \({S, T\subseteq \omega_1}\). A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of \({H_{\omega_2}}\) and the existence of sharps for all reals. Also, an optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin’s Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from \({H_{\omega_2}}\) implies that every function from ω1 into ω1 is bounded on a club by a canonical function.