Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for every finite n>0n>0 for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ  -uniform families in which ρ≥ℶω(ν)ρ≥ℶω(ν). The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.As corollaries, the use of additional axioms is eliminated from splitting theorems due to Erdős and Hajnal [1], Komjáth [7], Hajnal, Juhász and Shelah [4]; upper bounds are set on conflict-free coloring numbers of families of sets; and a general comparison theorem for ρ  -uniform families of sets is proved, which generalizes Komjáth's comparison theorem for ℵ0ℵ0-uniform families [8].