Splitting families of sets in ZFC

Abstract

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. 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.