Three forms of the Erd\H{o}s-Dushnik-Miller Theorem

TL;DR

This paper explores three distinct forms of the Erdős-Dushnik-Miller theorem within set theory lacking the axiom of choice, analyzing their logical strength and hierarchical positions.

Contribution

It identifies three inequivalent versions of the theorem and examines their placement in the hierarchy of weak choice principles.

Findings

Three inequivalent versions of the Erdős-Dushnik-Miller theorem identified.

Results on the relative strength of these versions in weak choice hierarchies.

Insights into the logical foundations of combinatorial set theory without choice.

Abstract

We continue the study of the Erd\H{o}s-Dushnik-Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positions of these versions in the deductive hierarchy of weak choice principles.