## Abstract

Miller's 1937 splitting theorem was proved for every finite n>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≥Beth_{ω}(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 Erdo's 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 ℵ_{0}-uniform families [8].

Original language | English |
---|---|

Pages (from-to) | 707-725 |

Number of pages | 19 |

Journal | Advances in Mathematics |

Volume | 269 |

DOIs | |

State | Published - 1 Jan 2015 |

## Keywords

- Conflict-free number
- Disjoint refinement
- Essentially disjoint family
- Family of sets
- Filtrations
- Generalized continuum hypothesis
- Löwenheim-Skolem theorem
- Miller's theorem
- Property B
- Shelah's revised GCH
- ZFC

## ASJC Scopus subject areas

- General Mathematics