  Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$
in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem
is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is
arbitrary and $\rho\ge \beth_\om(\nu)$. The proof uses a new general method
that is based on Shelah's revises Generalized Continuum Hypothesis theorem.
Upper bounds on conflict-free coloring numbers of families of sets and a
general comparison theorem follow as corollaries of the main theorem. Other
corollaries eliminate the use of additional axioms from splitting theorems due
to Erdos, Hajnal, Komjath, Juhasz and Shelah.
