  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 $n\in \mathbb N$.
