Rowbottom cardinal

In set theory, a Rowbottom cardinal (named after Frederick Rowbottom) is a certain kind of large cardinal number.

An uncountable cardinal number &kappa; is said to be "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 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).

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