Infinite-Exponent Partition Relations on Higher Analogues of the Real Line

TL;DR

This paper investigates infinite-exponent partition relations on higher-dimensional analogues of the real line, extending classical results to more complex linear orders within ZF set theory.

Contribution

It generalizes partition relations to linear orders of the form ${}^eta 2$ for ordinals $eta$, providing a full classification for countable $ au$ without relying on the Axiom of Choice.

Findings

Classified the relation ${}^eta 2 <_{lex} \rightarrow (\tau)^\tau$ for countable \tau.

Extended classical partition relation results to higher-order linear orders.

Operated entirely within ZF, avoiding the Axiom of Choice.

Abstract

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^\alpha 2,<_{\text{lex}}\rangle$ for $\alpha$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $\langle {}^\alpha 2,<_{\text{lex}}\rangle \rightarrow (\tau)^\tau$ for $\tau$ countable.