Halin's end degree conjecture

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.