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On ends of degree $ω_1$

Leandro Aurichi, Gabriel Fernandes, Paulo Magalhães Júnior

Abstract

We prove that if $ T $ is a semi-special tree that is not special, then there exists a graph $ G $, formed as an inflation of a sparse $ T $-graph, such that for any special tree $ S $, $ G $ is not a subdivision of an inflation of an sparse $ S $-graph. Furthermore $G$ has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin's grid theorem, extending it to ends of degree $ \aleph_1 $, which was originally established for ends of countable degree.

On ends of degree $ω_1$

Abstract

We prove that if is a semi-special tree that is not special, then there exists a graph , formed as an inflation of a sparse -graph, such that for any special tree , is not a subdivision of an inflation of an sparse -graph. Furthermore has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin's grid theorem, extending it to ends of degree , which was originally established for ends of countable degree.

Paper Structure

This paper contains 6 sections, 15 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. A different counter example for ends of degree
  4. Ends of degree
  5. Acknowledgments
  6. The first named author thanks the support of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), being sponsored through grant number 2023/00595-6. The third named author acknowledges the support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) through grant number 165761/2021-0.

Key Result

Theorem 1.1

Every graph with an end of infinite degree contains a subdivision of the hexagonal quarter grid whose rays belong to that end.

Theorems & Definitions (49)

  • Theorem 1.1: Halin's grid theorem - Halintheorem
  • Conjecture 1.2: Halin's Degree Conjecture - Halinconjecture, Conjecture 6.1
  • Theorem 1.4
  • Definition 1.5: The $HC^*$ property
  • Theorem 1.6
  • Definition 2.1: Order Trees
  • Definition 2.2: Accumulation points
  • Definition 2.3: Clubs, Stationary sets and Diamond Sequences
  • Lemma 2.4: Fodor's Lemma, Pressing Down
  • Definition 2.5
  • ...and 39 more