Rowbottom cardinal

In set theory, a Rowbottom cardinal, introduced by Rowbottom (1971), is a certain kind of large cardinal number.

An uncountable cardinal number ${\displaystyle \kappa }$ is said to be ${\displaystyle \lambda }$-Rowbottom if for every function f: [κ]^<ω → λ (where λ < κ) there is a set H of order type ${\displaystyle \kappa }$ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has < ${\displaystyle \lambda }$ elements. ${\displaystyle \kappa }$ is Rowbottom if it is ${\displaystyle \omega _{1}}$ - 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 + “${\displaystyle \aleph _{\omega }}$ 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 ${\displaystyle \aleph _{\omega }}$ is Rowbottom (but contradicts the axiom of choice).