= Normal ultrafilter on a cardinal
{title2=$\Delta_{\alpha<\kappa}X_\alpha\in U$}
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.
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