Structural Infinite-Exponent Partition Relations and Weak Choice Principles

TL;DR

This paper explores how certain infinite-exponent partition relations on structures like linear orders and graphs relate to weak choice principles, showing some are consistent with ZF and imply the failure of multiple choice axioms.

Contribution

It demonstrates the consistency of specific infinite-exponent partition relations with ZF and their implications for weak choice principles such as KWP$_1$ and O.

Findings

Some infinite-exponent partition relations are consistent with ZF.

These relations imply the failure of the Axiom of Choice.

They also imply the failure of Kinna-Wagner Selection Principle KWP$_1$ and the Ordering Principle O.

Abstract

We investigate infinite-exponent partition relations on arbitrary relational structures, with a focus on linear orders and graphs. Any such relation contradicts the Axiom of Choice. We show that there are some such relations which are consistent with ZF which imply the failure not just of Choice but also of the Kinna-Wagner Selection Principle KWP$_1$ and the Ordering Principle O.