Rowbottom cardinal

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

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

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).

References

Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3.
Rowbottom, Frederick (1971) [1964], "Some strong axioms of infinity incompatible with the axiom of constructibility", Annals of Pure and Applied Logic 3 (1): 1–44, doi:10.1016/0003-4843(71)90009-X, ISSN 0168-0072, MR 0323572