Splitting families of sets in ZFC

TL;DR

This paper extends Miller's splitting theorem within ZFC to broader cardinal pairs using a novel method based on Shelah's GCH theorem, and derives new bounds on conflict-free coloring numbers.

Contribution

It provides a general extension of Miller's splitting theorem for arbitrary and large cardinals within ZFC, introducing a new proof technique and removing reliance on additional axioms.

Findings

Extended Miller's splitting theorem to arbitrary cardinals and large bounds.

Derived upper bounds on conflict-free coloring numbers.

Eliminated need for extra axioms in related splitting theorems.

Abstract

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