A Simple Proof and Some Difficult Examples for Hindman's Theorem

TL;DR

This paper provides a straightforward proof of Hindman's Theorem and presents examples illustrating the computational complexity of finding witnesses in certain colorings.

Contribution

It introduces a simple, explicit proof of Hindman's Theorem and demonstrates the existence of colorings without computable witnesses, highlighting computational limitations.

Findings

Explicit proof of Hindman's Theorem

Examples of colorings lacking computable witnesses

Insights into computational complexity of the theorem

Abstract

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.