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 ${♢}_{{\omega }_1}$, 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 ${♢}_{{\omega }_1}^{\ast }$, 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.