Infinite-Exponent Partition Relations on the Real Line

TL;DR

This paper extends the theory of infinite-exponent partition relations to all linear order types, especially the real line, providing a complete classification and working within ZF without the Axiom of Choice.

Contribution

It offers a comprehensive classification of partition relations on the real line with countably infinite exponents, advancing the understanding of these relations in set theory.

Findings

Complete classification of partition relations on the real line

Characterization of the absence of uncountable-exponent relations

Work conducted in ZF without the Axiom of Choice

Abstract

We extend the theory of infinite-exponent partition relations to arbitrary linear order types, with a particular focus on the real number line. We give a complete classification of all consistent partition relations on the real line with countably infinite exponents, and a characterisation of the statement "no uncountable-exponent partition relations hold on the real line", working throughout in ZF without the Axiom of Choice.