Rowbottom cardinal

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

An uncountable cardinal number $$\kappa$$ is said to be $$\lambda$$-Rowbottom if for every function f: [&kappa;]<&omega; &rarr; &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 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 + “$$\aleph_{\omega}$$ 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 $$\aleph_{\omega}$$ is Rowbottom (but contradicts the axiom of choice).