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. Komjáth: 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 $n\in \mathbb{N}$.

8 pages, prepared for the Logic Colloquium 2014

References in corpus (2)

• Noetherian types of homogeneous compacta and dyadic compacta
• Trees, ladders and graphs