On ends of degree $ω_1$

Abstract: 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 $ \aleph_1 $, which was originally established for ends of countable degree.