A version of the Erdős-Rado theorem on partitions of the unordered n-tuples from uncountable sets is proved, without using the axiom of choice. The case with exponent 1 is just the Sierpinski-Hartogs’ result that ℵ(α) ≤ 222α.
"Erdős-Rado without choice." J. Symbolic Logic 72 (3) 897 - 900, September 2007. https://doi.org/10.2178/jsl/1191333846