Splitting families of sets in ZFC

Miller’s 1937 splitting theorem was proved for pairs of cardinals $(\text{n},\rho )$ 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 $\rho \ge {\mathrm{\beth }}_{\text{om}}(\nu )$. 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.