Rowbottom cardinal

Template:Short description In set theory, a Rowbottom cardinal, introduced by Frederick Rowbottom,Template:Sfnp is a certain kind of large cardinal number.

An uncountable cardinal number $\kappa$ is said to be Template:Math-Rowbottom if for every function $f:[\kappa {]}^{<\omega }\to \lambda$ (where $\lambda <\kappa$) 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 simply Rowbottom if it is Template:Math-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).

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