Abstract:
For an infinite cardinal λ, we say that λ is J´onsson if it satisfies the square bracket partition property λ → [λ]<ωλ . This means that, for everycoloring F : [λ]<ω → λ, there is some set H ∈ [λ]λ with the property that ran(F [H]<ω) ( λ. It is rather well known that the consistency strength of"there is a J´onsson cardinal" lies above "0] exists", but below the existence of a measurable. However, the question of what sorts of cardinals can be Jo´nsson has turned out to be rather difficult. In this talk, I will focus on the case where λ is inaccessible and present a result (due to Shelah), which tell us that if λ is an inaccessible Jo´nsson cardinal, then λ must be at least λ-Mahlo. Time permitting, I will discuss how one might hope to push this up to greatly Mahlo.