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

Arno Pauly

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.

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

Abstract

We study the complexity of the computational task ``Given a colouring , find a monochromatic such that ''. The framework is Weihrauch reducibility. Our results answer some open questions recently raised by Gill, and by Dzhafarov, Solomon and Valenti.
Paper Structure (6 sections, 32 theorems, 1 figure)

This paper contains 6 sections, 32 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Locating $\mathrm{TT}^1_k$
  4. Finite guessability
  5. Guessability and $\mathrm{TT}$
  6. The finitary part of $\mathrm{TT}^1$

Key Result

Proposition 1

$\mathrm{TT}^1_{k+1} \leq_{\textrm{W}} \mathrm{TC}_\mathbb{N}^k$

Figures (1)

  • Figure 1: All Weihrauch reductions (up to transitivity) between the principles $\mathrm{RT}^1_k$, $\mathrm{TT}^1_n$, $\mathrm{TC}_\mathbb{N}^j$ and $(\mathrm{RT}^1_\ell)'$.

Theorems & Definitions (60)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 4
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 50 more