A polarized partition relation using elementary substructures (Q2710592)

The following polarized partition relation is proved. Let \(\kappa\) be an infinite cardinal, \(\lambda=\bigl(2^{<\kappa}\bigr)^+\), \(\alpha<\lambda\) an ordinal. If the Cartesian product \(\lambda\times\lambda\) is colored with colors 0 and 1 then either there is in color 0 a homogeneous set \(A\times B\) with \(A\) of type \(\alpha\), \(B\) of type \(\lambda\), or vice versa, or else there is in color 1 a homogeneous set \(A\times B\) with \(A\) and \(B\) both of type \(\kappa+1\). This extends earlier results of Erdős, Hajnal, and Rado. The proof uses elementary submodels.