On Erdos--Dushnik--Miller theorem without AC

TL;DR

This paper explores the Erdos-Dushnik-Miller theorem within set theory lacking the Axiom of Choice, analyzing its placement in weak choice hierarchies and addressing a question by Soukup.

Contribution

It provides new insights into the theorem's validity without AC and clarifies its position in the hierarchy of weak choice principles.

Findings

The theorem's placement varies in weak choice contexts.

Partial answers to Soukup's question are provided.

The results deepen understanding of combinatorial set theory without AC.

Abstract

In set theory without the Axiom of Choice, we study the possible placement of Erdos-Dushnik-Miller theorem restricted to an uncountable set of vertices in the hierarchy of weak choice forms. We also answer a part of a question raised by Lajos Soukup.