Strong partition cardinal

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

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