Narrow systems revisited

We investigate connections between set‐theoretic compactness principles and cardinal arithmetic, introducing and studying generalized narrow system properties as a way to approach two open questions about two‐cardinal tree properties. The first of these questions asks whether the strong tree property at a regular cardinal implies the singular cardinals hypothesis () above . We show here that a certain narrow system property at that is closely related to the strong tree property, and holds in all known models thereof, suffices to imply above . The second of these questions asks whether the strong tree property can consistently hold simultaneously at all regular cardinals . 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.