Applications of Order Trees in Infinite Graphs

Abstract

Traditionally, the trees studied in infinite graphs are trees of height at most ω, with each node adjacent to its parent and its children (and every branch of the tree inducing a path or a ray). However, there is also a method, systematically introduced by Brochet and Diestel, of turning arbitrary well-founded order trees T into graphs, in a way such that every T-branch induces a generalised path in the sense of Rado. This article contains an introduction to this method and then surveys four recent applications of order trees to infinite graphs, with relevance for well-quasi orderings, Hadwiger’s conjecture, normal spanning trees and end-structure, the last two addressing long-standing open problems by Halin.