A partition relation for successors of Large Cardinals

Abstract

We address partition problems of Erdos and Hajnal by showing that $\kappa^+\to(\kappa^2+1,\alpha)$ for all $\alpha<\kappa^+$ , if $\kappa^{<\kappa}=\kappa$ and $\kappa$ carries a $\kappa$ -dense ideal. If $\kappa$ is measurable we show that $\kappa^+\to(\alpha)^2_n$ for $n<\omega, \alpha<\Omega$ where $\Omega$ is a very large ordinal less than $\kappa^+$ that is closed under all primitive recursive ordinal operations.