Indivisibility and uniform computational strength

Kenneth Gill

Published 2023-12-06Version 1

A countable structure is indivisible if for every coloring with finite range there is a monochromatic isomorphic subcopy of the structure. Each indivisible structure $\mathcal{S}$ naturally corresponds to an indivisibility problem $\mathsf{Ind}\ \mathcal{S}$, 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 $\mathsf{Ind}\ \mathbb{Q}$ and $\mathsf{Ind}\ \mathscr{E}$ from several benchmark problems, showing in particular that $\mathsf{C}_\mathbb{N} \vert_\mathrm{W} \mathsf{Ind}\ \mathbb{Q}$ and hence $\mathsf{Ind}\ \mathbb{Q}$ is strictly weaker than the problem of finding an interval in which some color is dense for a given coloring of $\mathbb{Q}$; and that the Weihrauch degree of $\mathsf{Ind}\ \mathscr{E}_k$ is strictly between those of $\mathsf{SRT}^2_k$ and $\mathsf{RT}^2_k$, where $\mathsf{Ind}\ \mathcal{S}_k$ is the restriction of $\mathsf{Ind}\ \mathcal{S}$ to $k$-colorings.