Rowbottom cardinal

In set theory, a Rowbottom cardinal, introduced by Template:Harvs, is a certain kind of large cardinal number.

An uncountable cardinal number $\kappa$ is said to be $\lambda$-Rowbottom if for every function f: [κ]^<ω → λ (where λ < κ) there is a set H of order type $\kappa$ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has < $\lambda$ elements. $\kappa$ is Rowbottom if it is ${\omega }_1$ - Rowbottom.

Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.

In general, Rowbottom cardinals need not be large cardinals in the usual sense: Rowbottom cardinals could be singular. It is an open question whether ZFC + “${\mathrm{\aleph }}_{\omega }$ is Rowbottom” is consistent. If it is, it has much higher consistency strength than the existence of a Rowbottom cardinal. The axiom of determinacy does imply that ${\mathrm{\aleph }}_{\omega }$ is Rowbottom (but contradicts the axiom of choice).

References

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