The Galvin-Prikry Theorem in the Weihrauch lattice

TL;DR

This paper classifies the computational complexity of the Galvin-Prikry theorem's fragments within the Weihrauch lattice, revealing their precise position relative to hyperjump operators and extending analysis to transfinite Borel levels.

Contribution

It provides a detailed Weihrauch degree classification of Galvin-Prikry theorem fragments, connecting them with hyperjump iterations and extending to transfinite Borel hierarchy levels.

Findings

Galvin-Prikry fragments lie between hyperjump iterations $ ext{HJ}^{n+1}$ and $ ext{HJ}^n$

A Turing jump ideal containing homogeneous sets must include the n-th hyperjump of X

Extended analysis to transfinite Borel hierarchy levels

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 $\Pi^1_1$-$\mathsf{CA}_0$ in the Weihrauch lattice. To establish this classification we obtain the following computability theoretic result: a Turing jump ideal containing homogeneous sets for all $\Delta^0_{n+1}(X)$ 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.