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

TL;DR

This paper explores the relationships between maximal independent sets, chain/antichain principles, and cofinal subsets in set theory without the Axiom of Choice, revealing new connections with weak choice principles.

Contribution

It establishes new results linking graph theory and order theory concepts to weak choice principles in ZF set theory.

Findings

Every locally finite connected graph has a maximal independent set.

Locally countable connected graphs also have maximal independent sets.

Partially ordered sets with finite antichains and regular chain sizes have predictable sizes.

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 $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ if $\aleph_{\alpha}$ 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 $v\in V_{G}$.