We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...) Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”. (shrink)

In this paper, we show that certain choiceless models of ZF originally constructed using an almost huge cardinal can be constructed using cardinals strictly weaker in consistency strength.

Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal $\kappa$ indestructible under $\kappa$-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly (...) compact cardinals are the elements of K or their measurable limit points, every $\kappa \in K$ is a supercompact cardinal indestructible under $\kappa$-directed closed forcing, and every $\kappa$ a measurable limit point of K is a strongly compact cardinal indestructible under $\kappa$-directed closed forcing not changing $\wp$. We then derive as a corollary a model for the existence of a strongly compact cardinal $\kappa$ which is not $\kappa^+$ supercompact but which is indestructible under $\kappa$-directed closed forcing not changing $\wp and remains non-$\kappa^+$ supercompact after such a forcing has been done. (shrink)

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.