On some variants of the club principle (Q2024087)

The club principle \(\clubsuit\), introduced by \textit{A. J. Ostaszewski} [J. Lond. Math. Soc., II. Ser. 14, 505--516 (1976; Zbl 0348.54014)], says that there exists a sequence \(\overline{A}=\left\langle A_\delta:\delta\in \text{Lim}(\omega_1)\right\rangle\) (where \(\text{Lim}(\omega_1)\) is the set of all countable limit ordinals), where each \(A_\delta\) is an unbounded subset of \(\delta\) of order type \(\omega\), such that for every \(A\in[\omega_1]^{\aleph_1}\) there exists \(\delta\in \text{Lim}(\omega_1)\) such that \(A_\delta\subseteq A\). In the paper under review, the authors study asymptotic versions of the club principle where the requirement \(A_\delta\subseteq A\) is replaced by \(A_\delta\cap A\) is a ``large'' subset of \(A_\delta\), continuing their previous work [Math. Log. Q. 64, No. 1--2, 44--48 (2018; Zbl 07198300)]. Using forcing technique, they prove that different versions of investigated principles are consistently non-equivalent.