Normal ultrafilter on a cardinal (${\Delta }_{\alpha <\kappa }{X}_{\alpha }\in U$)

A uniform kappa-complete ultrafilter $U$ on an uncountable cardinal $\kappa$ is normal if every regressive function on a $U$-large set is constant on a $U$-large subset. Equivalently it is closed under diagonal intersections ${\Delta }_{\alpha <\kappa }{X}_{\alpha }=\{\beta <\kappa :(\forall \alpha <\beta )\beta \in X_{\alpha }\}$. Normality supplies homogeneous sets for pair-colourings by combining the chosen large tail at each earlier coordinate.