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.