Essentially disjoint families, conflict free colorings and Shelah's Revised GCH

TL;DR

This paper proves new results on essentially disjoint families and conflict-free colorings of set families using Shelah's revised GCH theorem, including a compactness theorem for singular cardinals.

Contribution

It introduces novel theorems linking Shelah's revised GCH to properties of almost disjoint families and conflict-free colorings, extending previous combinatorial set theory results.

Findings

Every mu-almost disjoint family of subsets with size >= beth_omega is essentially disjoint.

Such families have conflict-free colorings with beth_omega colors.

A new compactness theorem for singular cardinals is established.

Abstract

Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Putting together these results we obtain that if mu<beth_omega<=lambda, then every mu-almost disjoint family B of subsets of lambda with |b|>=beth_omega for all b from B has a conflict-free colorings with beth_omega colors. To yield the above mentioned results we also need to prove a certain compactness theorem concerning singular cardinals.