Strong colorings based on oscillations

Abstract

We show that for any uncountable cardinal $κ$, there is a coloring $c: [κ]^2\to ω$ such that $c''A \otimes B = ω$ for any $A, B\subseteq κ$ of order type $ω_1$ that are stationary in their common supremum. In particular, the stationary version of Erdős-Rado theorem and the higher dimensional Friedman's property are both inconsistent. We demonstrate that the theorem is optimal in various ways.