# Blog Archives

# Topic Archive: singular cardinals

# Jónsson cardinals and club guessing

We say that a cardinal $\lambda$ is a Jónsson cardinal if it satisfies the following weak Ramsey-type property: given any coloring $F:[\lambda]^{<\omega}\to \lambda$ of the finite subsets of $\lambda$ in $\lambda$-many colors, there exists a set $H\in[\lambda]^\lambda$ such that the range of $F\upharpoonright [H]^{<\omega}$ is a proper subset of $\lambda$. One of the big driving forces present in early chapters Cardinal Arithmetic is an attempt to understand the combinatorial structure at and around Jónsson cardinals using scales and club guessing. The goal of this talk is to highlight the connection between Jónsson cardinals and the existence of certain sorts of club guessing ideals. Our focus will be on how club guessing ideals interact with Jónssonness at successors of singulars.