Davies-trees in infinite combinatorics

2014/07/14 by Daniel T. Soukup, Soukup, Daniel T.
Mathematics · #03C98 #03E05 #05C63 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #math.CO #math.LO #msc:03C98 #msc:03E05 #msc:05C63

paper · pdf · doi:10.48550/arxiv.1407.3604

8 pages, prepared for the Logic Colloquium 2014

arxiv created 2014/07/14 · arxiv updated 2014/07/15

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∈ \mathbb N; \mathbb R² is the union of n+2 clouds if the continuum is at most ℵ_n for any n∈ \mathbb N; every uncountably chromatic graph contains n-connected uncountably chromatic subgraphs for every n∈ \mathbb N.

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