Indivisibility and uniform computational strength

TL;DR

This paper studies the computational complexity of indivisibility problems for certain countable structures, revealing their position within the Weihrauch degrees and their relation to well-known combinatorial principles.

Contribution

It analyzes the Weihrauch degrees of indivisibility problems for the rational order and an equivalence relation, establishing their separation from benchmarks and their placement among Ramsey-type principles.

Findings

Indivisibility problem for $\

$ cannot solve the problem of finding a monochromatic rational interval.

The Weihrauch degree of the indivisibility problem for $$ is between ^2 and ^2, a variant of Ramsey's theorem.

Abstract

A countable structure is indivisible if for every coloring with finite range there is a monochromatic isomorphic subcopy of the structure. Each indivisible structure naturally corresponds to an indivisibility problem which outputs such a subcopy given a presentation and coloring. We investigate the Weihrauch complexity of the indivisibility problems for two structures: the rational numbers $\mathbb{Q}$ as a linear order, and the equivalence relation $\mathscr{E}$ with countably many equivalence classes each having countably many members. We separate the Weihrauch degrees of both corresponding indivisibility problems from several benchmarks, showing in particular that the indivisibility problem for $\mathbb{Q}$ cannot solve the problem of finding a monochromatic rational interval given a coloring for which there is one; and that the Weihrauch degree of the indivisibility problem for $\mathscr{E}$ is strictly between those of $\mathsf{RT}^2$ and $\mathsf{SRT}^2$, two widely studied variants of Ramsey's theorem for pairs whose reverse-mathematical separation was open until recently.