Ray inflations of $\omega_1$-trees and ends of degree $\aleph_1$

Abstract

We prove the followings result for ray inflations of sparse graphs on $\omega_1$-trees. First, let $T$ and $S$ be pruned $\omega_1$-trees, let $G_T$ be a sparse $T$-graph, and let $G_S$ be a sparse $S$-graph with uniformly finite adhesion. If $T$ is not special, then no subdivision of $G_S\# \mathbb{N}$ is isomorphic to $G_T\# \mathbb{N}$. Second, if $T$ is almost-Suslin and $S$ is special, then $G_T\# \mathbb{N}$ contains no subgraph isomorphic to $G_S\# \mathbb{N}$, for arbitrary choices of the sparse graphs. Consequently, under $\diamondsuit_{\omega_1}$, this gives a counterexample to Halin's end degree conjecture that is not isomorphic to a subdivision of any ray inflation with uniformly finite adhesion. Under $\diamondsuit_{\omega_1}^{*}$, the underlying tree may in addition be chosen almost-Suslin, and the resulting graph contains no subgraph isomorphic to a ray inflation over a special $\omega_1$-tree.