# New bounds on the strength of some restrictions of Hindman's Theorem

## TL;DR

This paper establishes bounds on the logical strength of certain restrictions of Hindman's Theorem, revealing that even simplified versions imply strong systems like ACA_0, and introduces the concept of apartness to analyze solution sets.

## Contribution

It provides new upper and lower bounds for restricted Hindman's Theorem variants and introduces the apartness condition to study solution set properties.

## Key findings

- Hindman's Theorem for sums of length ≤ 2 and 4 colors implies ACA_0.
- Restricted versions with simple proofs have better computability bounds.
- The apartness condition plays a key role in analyzing solution sets.

## Abstract

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies $\mathsf{ACA}_0$. An emerging {\em leitmotiv} is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.