In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number.
An uncountable cardinal number is said to be -Rowbottom if for every function f: [κ]<sup><ω</sup> → λ (where λ < κ) there is a set H of order type that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has < elements. is Rowbottom if it is - Rowbottom.
Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + âÂÂthere is a Rowbottom cardinalâ and ZFC + âÂÂthere is a Jónsson cardinalâ 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 + â is Rowbottomâ 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 is Rowbottom (but contradicts the axiom of choice).