Combinatorial principles on \(\omega_{1}\), cardinal invariants of the meager ideal and destructible gaps (Q2583064)

Say an \((\omega_1 , \omega_1)\)-gap \(({\mathcal A}, {\mathcal B})\) is destructible if there is an \(\aleph_1\)-preserving forcing extension in which \(({\mathcal A}, {\mathcal B})\) is not a gap. Otherwise \(({\mathcal A}, {\mathcal B})\) is called indestructible. Indestructible gaps exist while the existence of destructible gaps is independent of ZFC. Ostaszewski's \(\clubsuit\)-principle is the statement that there is a sequence \(\langle A_\alpha : \text{cf} (\alpha) = \omega\) and \(\alpha < \omega_1 \rangle\) with \(A_\alpha\) being a cofinal subset of \(\alpha\) such that for all uncountable \(B \subseteq \omega_1\), \(A_\alpha \subseteq B\) holds for stationarily many \(\alpha\)'s. \(\clubsuit\) is a weakening of Jensen's classical \(\diamondsuit\)-principle. The principle \(^{\bullet}*\) says there is family \({\mathcal A}\) of countable subsets of \(\omega_1\) of size \(\aleph_1\) such that for all uncountable \(B \subseteq \omega_1\) there is \(A \in {\mathcal A}\) with \(A \subseteq B\). Clearly \(\clubsuit\) implies \(^{\bullet}*\). The author shows that if \(^{\bullet}*\) holds and the real line cannot be covered by \(\aleph_1\) many meager sets, then there is a destructible gap. This sharpens Todorčević's result saying that adding a Cohen real adds a destructible gap. He also obtains the same conclusion under the assumption that \(\clubsuit\) holds and the meager ideal is generated by \(\aleph_1\) many sets, thus strengthening Todorčević's theorem that \(\diamondsuit\) implies the existence of a destructible gap. This work thus pursues the analogy between Suslin trees and destructible gaps for it has been known that Suslin trees exist under either of these assumptions.