Solovay's Theorem states
If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets.
For regular uncountable successor $\kappa$, this follows from a result by Ulam [Kunen, p.79].
My questions are:
1) Can I apply the theorem for $\kappa=\mathfrak c =2^\omega$, when CH fails ($\mathfrak c >\omega_1$)? (is $\mathfrak c$ necessarily regular?).
2) Might I be able to prove the special case in (1) using methods less sophisticated than those found in Solovay's proof? Something like Ulam's matrix?