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The Galvin-Prikry Theorem in the Weihrauch lattice

Alberto Marcone, Gian Marco Osso

Abstract

This paper classifies different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey's theorem, in terms of their uniform computational content (Weihrauch degree). It can be seen as a continuation of arXiv:2003.04245v3, which focused on the Weihrauch classification of functions related to the open (and clopen) Ramsey theorem. We show that functions related to the Galvin-Prikry theorem for Borel sets of rank n are strictly between the (n+1)-th and n-th iterate of the hyperjump operator $\mathsf{HJ}$, which is in turn equivalent to the better known $\widehat{\mathsf{WF}}$, which corresponds to $Π^1_1$-$\mathsf{CA}_0$ in the Weihrauch lattice. To establish this classification we obtain the following computability theoretic result: a Turing jump ideal containing homogeneous sets for all $Δ^0_{n+1}(X)$ sets must also contain the n-th hyperjump of X. We also extend our analysis to the transfinite levels of the Borel hierarchy. We further obtain some results about the reverse mathematics of the lightface fragments of the Galvin-Prikry theorem.

The Galvin-Prikry Theorem in the Weihrauch lattice

Abstract

This paper classifies different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey's theorem, in terms of their uniform computational content (Weihrauch degree). It can be seen as a continuation of arXiv:2003.04245v3, which focused on the Weihrauch classification of functions related to the open (and clopen) Ramsey theorem. We show that functions related to the Galvin-Prikry theorem for Borel sets of rank n are strictly between the (n+1)-th and n-th iterate of the hyperjump operator , which is in turn equivalent to the better known , which corresponds to - in the Weihrauch lattice. To establish this classification we obtain the following computability theoretic result: a Turing jump ideal containing homogeneous sets for all sets must also contain the n-th hyperjump of X. We also extend our analysis to the transfinite levels of the Borel hierarchy. We further obtain some results about the reverse mathematics of the lightface fragments of the Galvin-Prikry theorem.

Paper Structure

This paper contains 15 sections, 81 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic notions
  3. The Galvin-Prikry theorem and the Ramsey space
  4. Reverse mathematics
  5. Computable analysis
  6. The function $\beta\text{-}\mathsf{mod}$
  7. Trees, $\Pi^0_1$ classes, and the hyperjump
  8. Definition and Weihrauch degree of $\beta\text{-}\mathsf{mod}$
  9. Computing choice functions from $\beta\text{-}\mathsf{mod}$
  10. Degree theoretic analysis of the Galvin-Prikry theorem
  11. The formula $\varphi$ and its properties
  12. The finite Borel levels
  13. The transfinite Borel levels
  14. Weihrauch classification
  15. Reverse mathematics of the lightface Galvin-Prikry theorem

Key Result

Theorem 2.1

Let $[\omega]^{\omega} =A_0 \cup A_1$ be a partition of $[\omega]^{\omega}$ into Borel sets. There exists some $H \subseteq \omega$ which is homogeneous for the partition, i.e. such that $[H]^{\omega} \subseteq A_i$ for some $i < 2$.

Figures (2)

  • Figure 1: This table is a picture of the arithmetical Weihrauch degrees of functions related to the Galvin-Prikry theorem for Borel sets of rank $k+1$. A solid black arrow from $f$ to $g$ means $f <^{\mathrm{a}}_{\mathrm{W}} g$, while a dashed one denotes $f \leq^{\mathrm{a}}_{\mathrm{W}} g$ (and $g\leq^{\mathrm{a}}_{\mathrm{W}} f$ is open) and a red one indicates $f \nleq^{\mathrm{a}}_{\mathrm{W}} g$. Labeled arrows reference the proofs of the reductions and/or separations. Unlabeled arrows correspond to trivial reductions and/or separations.
  • Figure 2: This table is a picture of the Weihrauch degrees of functions related to the Galvin-Prikry theorem for Borel sets of rank $k$. A solid black arrow from $f$ to $g$ indicates $f <_{\mathrm{W}} g$, while a dashed one indicates $f \leq_{\mathrm{W}} g$ (and $g\leq_{\mathrm{W}} f$ is open) and a red one indicates $f \nleq_{\mathrm{W}} g$. Labeled arrows reference the proofs of the reductions and/or separations. Unlabeled arrows correspond to trivial reductions and/or separations.

Theorems & Definitions (169)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6: simpson
  • Lemma 2.7: simpson
  • ...and 159 more