Jónsson cardinal

In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.

An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ]^<ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ.

Every Rowbottom cardinal is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. William Mitchell proved, with the help of the Dodd-Jensen core model that the consistency of the existence of a Jónsson cardinal implies the consistency of the existence of a Ramsey cardinal, so that the existence of Jónsson cardinals and the existence of Ramsey cardinals are equiconsistent.

In general, Jónsson cardinals need not be large cardinals in the usual sense: they can be singular. But the existence of a singular Jónsson cardinal is equiconsistent to the existence of a measurable cardinal. Using the axiom of choice, a lot of small cardinals (the ${\displaystyle \aleph _{n}}$, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: The axiom of determinacy does imply that for every positive natural number n, the cardinal ${\displaystyle \aleph _{n}}$ is Jónsson.

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