Davies-trees in infinite combinatorics

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 $n\in \mathbb N$.