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

• Henle, James M.; Kleinberg, Eugene M.; Watro, Ronald J. (1984), "On the ultrafilters and ultrapowers of strong partition cardinals", Journal of Symbolic Logic, 49 (4): 1268–1272, doi:10.2307/2274277, JSTOR 2274277, S2CID 45989875
• Apter, Arthur W.; Henle, James M.; Jackson, Stephen C. (1999), "The calculus of partition sequences, changing cofinalities, and a question of Woodin", Transactions of the American Mathematical Society, 352 (3): 969–1003, doi:10.1090/S0002-9947-99-02554-4, JSTOR 118097, MR 1695015.