Working in ZFC, we show that for any infinite cardinal $\kappa$ and ordinal $\gamma < (2^{<\kappa})^+$ the polarized partition relation $\[\begin{pmatrix} (2^{<\kappa})^+\\ (2^{<\kappa})^+ \end{pmatrix}\]$ $\rightarrow$ $\[\begin{pmatrix}(2^{<\kappa})^+ & \gamma & \kappa + 1 \\ \vee \\ \gamma & (2^{<\kappa})^+ & \kappa + 1 \end{pmatrix}^{1,1}\]$ holds. Our proof of this relation involves the use of elementary substructures of set models of large fragments of ZFC.
@article{1183746248,
author = {Jones, Albin L.},
title = {A Polarized Partition Relation Using Elementary Substructures},
journal = {J. Symbolic Logic},
volume = {65},
number = {1},
year = {2000},
pages = { 1491-1498},
language = {en},
url = {http://dml.mathdoc.fr/item/1183746248}
}
Jones, Albin L. A Polarized Partition Relation Using Elementary Substructures. J. Symbolic Logic, Tome 65 (2000) no. 1, pp. 1491-1498. http://gdmltest.u-ga.fr/item/1183746248/