Another \(\diamondsuit\)-like principle (Q2717682)

The author introduces a~new combinatorial principle~\(\diamondsuit_{\mathfrak d}\). It states that there is a~\(\diamondsuit_{\mathfrak d}\)-sequence, i.e., a~sequence of functions \(d_\alpha:\alpha\to\omega\) such that for every \(f:\omega_1\to\omega\) there is \(\alpha\geq\omega\) such that \(f{\restriction}\alpha\leq^*d_\alpha\). This principle is a~slight strengthenning of the assumption \(\mathfrak d=\omega_1\). The author proves that \(\diamondsuit_{\mathfrak d}\)~implies \(\mathfrak a=\omega_1\), and \(\diamondsuit_{\mathfrak d}\)~follows from the conjunction of \(\mathfrak d=\omega_1\) and the \(\clubsuit\)~principle. On the other hand he proves that the theories \(\text{CH}+\neg\diamondsuit_{\mathfrak d}\), \(\neg\text{CH}+\diamondsuit_{\mathfrak d}\), \(\text{CH}+\diamondsuit_{\mathfrak d}+ \neg\diamondsuit\) are consistent. The principle~\(\diamondsuit_{\mathfrak d}\) is shown to be true in the extension by a~single Laver real while the extensions by measure algebras preserve both \(\diamondsuit_{\mathfrak d}\) and~\(\neg\diamondsuit_{\mathfrak d}\).