# 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.

## References

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