# Daniel Soukup: Davies-trees in infinite combinatorics

28 February 2014, 13:30–15:00
Fields institute, Room 210

Speaker:  Daniel Soukup

Title:  Davies-trees in infinite combinatorics

Abstract:

The aim of this talk is to introduce Davies-trees and present new applications to combinatorics. Davies-trees are special sequences of countable elementary submodels which played important roles in generalizing arguments using CH to pure ZFC proofs. My goal is to present two unrelated but fascinating results due to P. Komjáth: we prove that the plane is the union of n+2 “clouds” provided that the continuum is at most $\aleph_n$ and that every uncountably chromatic graph contains k-connected uncountably chromatic subgraphs for each finite k. We hopefully have time to review the most important open problems around the second theorem.