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 grid theorem. In particular, every end of regular uncountable degree κ would contain a star of rays, i.e., a configuration consisting of a central ray R and κ neighbouring rays (Ri: i < κ) all disjoint from each other and each Ri 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 1א, holds for \({\aleph _2},{\aleph _3}, \ldots ,{\aleph _\omega}\) fails for אω+1, and is undecidable (in ZFC) for the next אω+n with n ∈ ℕ, n ≽ 2. Further results include a complete solution for all cardinals under GCH, complemented by a number of consistency results.
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References
N. Bowler, J. Carmesin, P. Komjáth and C. Reiher, The colouring number of infinite graphs, Combinatorica 39 (2019), 1225–1235.
N. Bowler, S. Geschke and M. Pitz, Minimal obstructions for normal spanning trees, Fundamenta Mathematicae 241 (2018), 245–263.
J.-M. Brochet and R. Diestel, Normal tree orders for infinite graphs, Transactions of the American Mathematical Society 345 (1994), 871–895.
R. Diestel, A short proof of Halin’s grid theorem, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 74 (2004), 237–242.
R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173, Springer, Berlin, 2017.
R. Diestel and I. Leader, Normal spanning trees, Aronszajn trees and excluded minors, Journal of the London Mathematical Society 63 (2001), 16–32.
M. Gitik and M. Magidor, The singular cardinal hypothesis revisited, in Set Theory of the Continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, Vol. 26, Springer, New York, 1992, pp. 243–279.
J. P. Gollin and K. Heuer, Characterising k-connected sets in infinite graphs, Journal of Combinatorial Theory. Series B 157 (2022), 451–499.
R. Halin, Über die Maximalzahl fremder unendlicher Wege in Graphen, Mathematische Nachrichten 30 (1965), 63–85.
R. Halin, Miscellaneous problems on infinite graphs, Journal of Graph Theory 35 (2000), 128–151.
T. Jech, Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003.
H. A. Jung, Zusammenzüge und Unterteilungen von Graphen, Mathematische Nachrichten 35 (1967), 241–267.
M. Pitz, Proof of Halin’s normal spanning tree conjecture, Israel Journal of Mathematics 246 (2021), 353–370.
S. Shelah, Cardinal Arithmetic, Oxford Logic Guides, Vol. 29. The Clarendon Press, Oxford University Press, New York, 1994
S. Todorčević, Walks on Ordinals and Their Characteristics, Progress in Mathematics, Vol. 263, Birkhäuser, Basel, 2007.
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Geschke, S., Kurkofka, J., Melcher, R. et al. Halin’s end degree conjecture. Isr. J. Math. 253, 617–645 (2023). https://doi.org/10.1007/s11856-022-2368-5
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DOI: https://doi.org/10.1007/s11856-022-2368-5