Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

TL;DR

This paper explores the set-theoretic strength of various combinatorial and graph-theoretic statements without relying on the axiom of choice, revealing how certain properties hold or fail in such frameworks.

Contribution

It provides new insights into the validity of combinatorial principles and graph properties in set theory without the axiom of choice, including results on chromatic numbers and homomorphisms.

Findings

Certain chain and antichain properties imply countability or specific sizes without AC.

The chromatic number of product graphs behaves predictably under finite and infinite conditions.

Finite subgraph homomorphism properties extend to infinite graphs without AC.

Abstract

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.