Splitting families of sets in ZFC

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9 Scopus citations

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≥Bethω(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 ℵ0-uniform families [8].

Original languageEnglish
Pages (from-to)707-725
Number of pages19
JournalAdvances in Mathematics
Volume269
DOIs
StatePublished - 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

  • Mathematics (all)

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