Remarks on Halin's end-degree Conjecture

We prove new instances of Halin’s end degree conjecture (HC) in ZFC. In particular, we show that there is a proper class of cardinals kappa for which Halin’s conjecture holds, answering two questions posed by Geschke, Kurkofka, Melcher, and Pitz (2023). We also investigate the relationship between HC and the Singular Cardinal Hypothesis, deriving consistency strength from failures of the former. Moreover, we verify that Halin’s conjecture fails on finite intervals of successors of singular cardinals in Merimovich’s model, yielding a new independence result concerning HC.