1. Larger cardinalities
We have seen that (Ramsey) and (-Rado) for any On the other hand, we also have that () and (-Kakutani) for any infinite
Positive results can be obtained for larger cardinals than if we relax the requirements in some of the colors. A different extension, the -Rado theorem, will be discussed later.
This was originally shown by Dushnik and Miller in 1941 for regular, with providing the singular case. For regular one can in fact show something stronger:
Theorem 2 (-Rado) Suppose is regular and uncountable. Then
which means: If then either there is a stationary that is -homogeneous for , or else there is a closed subset of of order type that is -homogeneous for .
(Above, top stands for “topological.”)