On ends of degree ${\omega }_1$

We prove that if $T$ is a semi-special tree that is not special, then there exists a graph $G$, formed as an inflation of a sparse $T$-graph, such that for any special tree $S$, $G$ is not a subdivision of an inflation of an sparse $S$-graph. Furthermore $G$ has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin’s grid theorem, extending it to ends of degree ${\mathrm{\aleph }}_1$, which was originally established for ends of countable degree.