More on the indivisibility of $\mathbb{Q}$

TL;DR

This paper investigates the computational complexity of finding monochromatic dense linear orders within colorings of the rationals, using Weihrauch reducibility, and addresses open questions in the field.

Contribution

It provides new insights into the complexity of a classical combinatorial problem on the rationals within the Weihrauch framework, answering recent open questions.

Findings

Identifies the Weihrauch degree of the problem.

Clarifies the computational difficulty of finding monochromatic dense subsets.

Answers open questions posed by Gill, and Dzhafarov, Solomon, and Valenti.

Abstract

We study the complexity of the computational task ``Given a colouring $c : \mathbb{Q} \to \mathbf{k}$, find a monochromatic $S \subseteq \mathbb{Q}$ such that $(S,<) \cong (\mathbb{Q},<)$''. The framework is Weihrauch reducibility. Our results answer some open questions recently raised by Gill, and by Dzhafarov, Solomon and Valenti.