"Weak yet strong" restrictions of Hindman's Finite Sums Theorem

TL;DR

This paper introduces a natural restriction of Hindman's Finite Sums Theorem that balances simplicity and computational strength, providing new insights into its proof-theoretic and computability-theoretic boundaries.

Contribution

It presents a new restricted version of Hindman's Theorem with a simple combinatorial proof and high computational complexity, bridging a gap in understanding its lower bounds.

Findings

The restriction admits a simple combinatorial proof.

It implies the existence of the Turing Jump.

It is the first example of a restriction with these properties.

Abstract

We present a natural restriction of Hindman's Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman's Theorem with analogous properties.