Strong colorings based on oscillations

Abstract

We show that for any uncountable cardinal $\kappa$, there is a coloring $c:[\kappa {]}^2\to \omega$ such that $c^{\prime \prime }A\otimes B=\omega$ for any $A,B\subseteq \kappa$ of order type ${\omega }_1$ that are stationary in their common supremum. In particular, the stationary version of Erdős-Rado theorem and the higher dimensional Friedman's property are both inconsistent. We demonstrate that the theorem is optimal in various ways.

arXiv comment10 pages