Characterising $k$-connected sets in infinite graphs

Gollin, J. Pascal; Heuer, Karl

doi:10.48550/arxiv.1811.06411

Cited by 2 publications

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“…Finally, if all components are smaller than κ and their size does not form a cofinal sequence in κ, there are at least κ many components in which case we even find a star of size κ. See also [8,Corollary 8.1] for a strengthening of the above results.…”

mentioning

confidence: 63%

Halin's end degree conjecture

Geschke¹,

Kurkofka²,

Melcher³

et al. 2020

Preprint

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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 ℵ1, holds for ℵ2, ℵ3, . . . , ℵω, 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.2010 Mathematics Subject Classification. 05C63.

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