Remarks on Halin's end-degree Conjecture

TL;DR

This paper advances the understanding of Halin's end-degree conjecture by proving new instances in ZFC, exploring its relation to the Singular Cardinal Hypothesis, and demonstrating its failure in certain models, thus providing new independence results.

Contribution

It proves new instances of Halin's conjecture in ZFC, investigates its connection to the Singular Cardinal Hypothesis, and shows its failure in specific models, offering new independence results.

Findings

Halin's conjecture holds for a proper class of cardinals.

The relationship between HC and the Singular Cardinal Hypothesis is clarified.

Halin's conjecture fails on finite intervals of successors of singular cardinals in Merimovich's model.

Abstract

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.