Topological ubiquity of trees

Abstract

Let ⊲ be a relation between graphs. We say a graph G is ⊲-ubiquitous if whenever Γ is a graph with nG⊲Γ for all n∈N, then one also has ℵ0G⊲Γ, where αG is the disjoint union of α many copies of G. The Ubiquity Conjecture of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.