Narrow systems revisited

Abstract

Motivated by two open questions about two-cardinal tree properties, we introduce and study generalized narrow system properties. The first of these questions asks whether the strong tree property at a regular cardinal $\kappa \geq \omega_2$ implies the Singular Cardinals Hypothesis ($\mathsf{SCH}$) above $\kappa$. We show here that a certain narrow system property at $\kappa$ that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply $\mathsf{SCH}$ above $\kappa$. The second of these questions asks whether the strong tree property can consistenty hold simultaneously at all regular cardinals $\kappa \geq \omega_2$. We show here that the analogous question about the generalized narrow system property has a positive answer. We also highlight some connections between generalized narrow system properties and the existence of certain strongly unbounded subadditive colorings.