Strong colorings based on oscillations

We show that for any uncountable cardinal $\kappa$, there is a coloring $c: [\kappa]^2\to \omega$ such that $c''A \otimes B = \omega$ for any $A, B\subseteq \kappa$ of order type $\omega_1$ that are stationary in their common supremum. In particular, the stationary version of Erd\H{o}s-Rado theorem and the higher dimensional Friedman's property are both inconsistent. We demonstrate that the theorem is optimal in various ways.