Solovay’s stationary partition theorem

The nonzero limit ordinals below kappa form a club: above any bound iterate successors omega times to find a larger limit below kappa, and a nonzero limit of such ordinals is a limit. Intersect S with this club and the tail above omega. Partition the resulting stationary T into $T_0=\{\alpha \in T:\mathrm{cf}(\alpha )<\alpha \}$ and $T_1=\{\alpha \in T:\mathrm{cf}(\alpha )=\alpha \}$. At least one is stationary.

If $T_0$ is stationary, its cofinality map is regressive. Fodor supplies a stationary subset of one cofinality $\lambda <\kappa$, which is infinite regular because its arguments are limits. The fixed-cofinality splitting lemma partitions this subset.

If $T_1$ is stationary, its members are regular uncountable cardinals: cofinalities of limits are regular cardinals, and these members equal their cofinalities and exceed omega. Apply the regular-cardinal splitting lemma. In either case adjoin every discarded point of S to one of the kappa pieces. Supersets preserve stationarity, and the pieces remain disjoint and exhaust S.