3 problems

Consistency conjecture. The property $(*)$ is consistent with ZFC.

Let $V=L$. Let $\mathbb{P}$ be a ccc forcing notion and let $G\subseteq\mathbb{P}$ be generic. Write $\clubsuit_S$ for the clubsuit principle associated with a stationary set…

Let $\langle \kappa_n \mid n < \omega\rangle$ be a sequence of infinitely many measurable cardinals. For each $n$, suppose that $\kappa_n$ carries an $(\omega,\kappa_{n-1}^+)$ repe…