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 + “