The reverse mathematics of Carlson's theorem for located words

TL;DR

This paper proves Carlson's theorem for located words within the framework of reverse mathematics, offering combinatorial and topological dynamic proofs to deepen understanding of the theorem's foundations.

Contribution

It provides two novel proofs of Carlson's theorem for located words in ACA^+_0, one combinatorial and one using topological dynamics.

Findings

Two proofs of Carlson's theorem for located words in ACA^+_0

A combinatorial proof similar to Towsner's proof of Hindman's theorem

A topological dynamics proof linking bounded sums to Carlson's theorem

Abstract

In this article, we give two proofs of Carlson's theorem for located words in~$\mathsf{ACA}^+_0$. The first proof is purely combinatorial, in the style of Towsner's proof of Hindman's theorem. The second uses topological dynamics to show that an iterated version of Hindman's theorem for bounded sums implies Carlson's theorem for located words.