A weak variant of Hindman's Theorem stronger than Hilbert's Theorem

TL;DR

This paper introduces the Adjacent Hindman's Theorem, a natural variant of Hindman's Theorem that is provable from Ramsey's Theorem for pairs and is strictly stronger than a previously studied weaker variant, Hilbert's Theorem.

Contribution

The paper defines the Adjacent Hindman's Theorem, proves its strength relative to Ramsey's Theorem, and establishes it as a natural, stronger restriction of Hindman's Theorem.

Findings

Adjacent Hindman's Theorem is provable from Ramsey's Theorem for pairs.

It is strictly stronger than Hirst's Hilbert's Theorem.

The theorem is a natural restriction of Hindman's Theorem.

Abstract

Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over $\RCA_0$ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem -- which we name the Adjacent Hindman's Theorem -- and prove it to be provable from Ramsey's Theorem for pairs and strictly stronger than Hirst's Hilbert's Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman's Theorem to the Increasing Polarized Ramsey's Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman's Theorem homogeneity is required only for finite sums of adjacent elements.