Strong partition cardinal

In Zermelo-Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal k such that every partition of the set [ k ] k of size k subsets of k into less than k pieces has a homogeneous set of size k .

The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ₁ is a strong partition cardinal.