Halin's end degree conjecture

TL;DR

This paper investigates Halin's end degree conjecture in infinite graph theory, revealing its validity or failure for various uncountable degrees and establishing its undecidability in ZFC, with comprehensive results under GCH.

Contribution

It provides the first detailed analysis of Halin's conjecture across different uncountable degrees, including proofs of its failure, validity, and undecidability in ZFC, along with solutions under GCH.

Findings

Conjecture fails for end degree 

Conjecture holds for , , ..., 

Conjecture undecidable for + in ZFC

Abstract

An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in this equivalence class. Halin conjectured that the end degree can be characterised in terms of certain typical ray configurations, which would generalise his famous \emph{grid theorem}. In particular, every end of regular uncountable degree $\kappa$ would contain a \emph{star of rays}, i.e.\ a configuration consisting of a central ray $R$ and $\kappa$ neighbouring rays $(R_i \colon i < \kappa)$ all disjoint from each other and each $R_i$ sending a family of infinitely many disjoint paths to $R$ so that paths from distinct families only meet in $R$. We show that Halin's conjecture fails for end degree $ \aleph_1$, holds for $\aleph_2,\aleph_3,\ldots,\aleph_\omega$, fails for $ \aleph_{\omega+1}$, and is undecidable (in ZFC) for the next $\aleph_{\omega+n}$ with $n \in \mathbb{N}$, $n \geq 2$. Further results include a complete solution for all cardinals under GCH, complemented by a number of consistency results.