## 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.

An old question by Halin asks whether the end degree can be characterised in terms of typical ray configurations. Halin conjectured that it can – in a very strong form which would generalise his famous

We show that Halin’s conjecture fails for end degree ℵ

An old question by Halin asks whether the end degree can be characterised in terms of typical ray configurations. Halin conjectured that it can – in a very strong form 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 (R_{i}:i<κ) 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 ℵ

_{1}, holds for end degree ℵ_{2},ℵ_{3},…,ℵ_{ω}, fails for ℵ_{ω+1}, and is undecidable (in ZFC) for the next ℵ_{ω+n}with n∈ℕ, n⩾2 .Original language | English |
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Title of host publication | Extended Abstracts EuroComb 2021 |

Subtitle of host publication | European Conference on Combinatorics, Graph Theory and Applications |

Editors | Jaroslav Nešetřil, Guillem Perarnau, Juanjo Rué, Oriol Serra |

Publisher | Birkhauser |

Pages | 78–83 |

Number of pages | 6 |

Edition | 1 |

ISBN (Electronic) | 9783030838232 |

ISBN (Print) | 9783030838225 |

DOIs | |

Publication status | Published - 24 Aug 2021 |

Event | European Conference on Combinatorics, Graph Theory and Applications 2021 - Online, Barcelona , Spain Duration: 6 Sept 2021 → 10 Sept 2021 |

### Publication series

Name | Trends in Mathematics |
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Publisher | Springer |

Volume | 14 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Conference

Conference | European Conference on Combinatorics, Graph Theory and Applications 2021 |
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Abbreviated title | EUROCOMB'21 |

Country/Territory | Spain |

City | Barcelona |

Period | 6/09/21 → 10/09/21 |

## Keywords

- Infinite graph
- Ends
- End degree
- Ray graph