Davies-trees in infinite combinatorics

TL;DR

This paper introduces Davies-trees and demonstrates their utility in infinite combinatorics by providing new, simplified proofs for several key theorems related to set families, continuum, and chromatic graphs.

Contribution

It presents novel applications of Davies-trees in infinite combinatorics, offering simpler proofs for existing theorems by P. Komjáth.

Findings

Every n-almost disjoint family of sets is essentially disjoint.

^2 is the union of n+2 clouds if continuum e0 e4 \u00a7 e4 n.

Uncountably chromatic graphs contain n-connected uncountably chromatic subgraphs.

Abstract

This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komj\'ath: every $n$-almost disjoint family of sets is essentially disjoint for any $n\in \mathbb N$; $\mathbb R^2$ is the union of $n+2$ clouds if the continuum is at most $\aleph_n$ for any $n\in \mathbb N$; every uncountably chromatic graph contains $n$-connected uncountably chromatic subgraphs for every $n\in \mathbb N$.