Rowbottom cardinal

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

An uncountable cardinal number $\kappa$ is said to be λ-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 ω₁-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).

References

• Kanamori, Akihiro (2003) [1994]. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Berlin, Heidelberg: Springer. ISBN 3-540-00384-3.

• Rowbottom, Frederick (1971). "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic. 3 (1): 1–44. doi:10.1016/0003-4843(71)90009-X. MR 0323572.