Superclub, splitting, separating statements
Shimon Garti, Saharon Shelah
TL;DR
The paper investigates how diamond-like prediction principles relate to tiltan and the stronger superclub, and how these principles influence cardinal characteristics. It defines and compares $\Diamond_S$, $\clubsuit_S$, and $\clubsuit^\Diamond_S$, proving that $\clubsuit^\Diamond_{\kappa^+}$ implies Galvin's property and yields a separation from $\clubsuit_{\kappa^+}$; it also shows that, at $\aleph_1$, superclub implies $\mathfrak{s}=\aleph_1$, and extends to $\mathfrak{s}_\kappa=\kappa^+$ when $\kappa$ is weakly compact. The results include forcing and preservation lemmas (e.g., Prikry forcing) that enable separations between tiltan and superclub at successors of large cardinals, and demonstrate that $\clubsuit_{\kappa^+}$ can coexist with $\neg{\rm Gal}(\mathscr{D}_{\kappa^+},\kappa^+,\kappa^{++})$. Collectively, the work clarifies the hierarchy among prediction principles and their consequences for cardinal characteristics, offering several open problems about potential global separations and consistency constraints.
Abstract
We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_κ=κ^+$. Based on this statement, we separate tiltan from superclub at a successor of a supercompact cardinal. We use Galvin's property in order to separate tiltan from superclub at successors of both regular and singular cardinals.