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Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Amitayu Banerjee

Abstract

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $\aleph_α$, then the set has size $\aleph_α$ if $\aleph_α$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that $\{w < v : \{w,v\} \in E_{G}\}$ is a clique for each $v\in V_{G}$.

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Abstract

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size , then the set has size if is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If is a connected locally finite chordal graph, then there is an ordering of such that is a clique for each .

Paper Structure

This paper contains 22 sections, 18 theorems, 1 equation, 1 figure.

Table of Contents

  1. introduction
  2. Maximal independent sets
  3. A variant of Chain/Antichain principle
  4. Cofinal well-founded subsets and consistency results
  5. A generalized formulation of the $n$-coloring theorem
  6. Locally finite connected graphs
  7. Notations, definitions, and known results
  8. Group-theoretical facts
  9. Permutation models.
  10. Loeb's theorem
  11. A theorem of Fulkerson and Gross
  12. Graph theoretical observations
  13. Maximal independent set
  14. The graph homomorphism problem
  15. Locally finite connected graphs
  16. ...and 7 more sections

Key Result

Lemma 2.5

The following hold.

Figures (1)

  • Figure 1: The graph $G$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 32 more