2012/09/06 by Kojman, Menachem
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1209.1307
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≥ \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.