# Locally finite trees and the topological minor relation

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO (better-quasi-order). Naturally, two interesting questions arise:

1. What is the number \lambda of topological types of locally finite trees?

2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this talk we address both questions by showing – entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

This talk is part of the Topology and Dynamics seminar series.