Erdos-Rado without choice

A version of the Erdös-Rado theorem on partitions of the unordered ntuples 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) ≤ 2 2α . The liberal use made by Erdös and Rado in [2] of the cardinal arithmetic versions of the axiom of choice enables them to give their result a particularly simple expression. So simple, indeed, that the elegant construction underlying it is not brought to the fore. As it happens there is a nontrivial combinatorial theorem about uncountable monochromatic sets which this construction serves up, and it deserves to be isolated. The referee for this paper made some very helpful comments and these are recorded in an appendix to the online version of this paper at www.dpmms.cam.ac.uk/~tf/erdosrado.pdf. Arithmetic notations An aleph is a cardinal of a wellordered set. אα is the αth aleph, and in this usage α is of course an ordinal. א(α) is the least aleph 6≤ α, and in this usage α is a cardinal not an ordinal, and the Hebrew letter is being used to denote Hartogs’ aleph function. When α is itself an aleph we often write ‘α’ for ‘א(α)’. By abuse of notation we will often use a notation denoting an aleph, such as ‘א(α)’ or ‘אκ’ to denote also the corresponding initial ordinal. Finally i0(α) =: α; in+1(α) =: 2in(α). Combinatorial notations [X] is the set of unordered n-tuples from X. “α → (β)γδ” means: take a set A of size α, partition the unordered γ-tuples of it into δ bits. Then there is a subset B ⊆ A of size β such that all the unordered γ-tuples from it are in the same piece of the partition. Here γ will always be in IN, and α, β and δ will be infinite cardinals.