Hindman's Theorem is only a countable phenomenon

TL;DR

This paper investigates the possibility of extending Hindman's Theorem to uncountable cardinals and finds that such generalizations generally do not hold at uncountable levels, unlike Ramsey's Theorem.

Contribution

It demonstrates that natural extensions of Hindman's Theorem to uncountable cardinals are mostly false, contrasting with the successful generalization of Ramsey's Theorem.

Findings

Generalizations of Hindman's Theorem fail at all uncountable cardinals.

Contrast with Ramsey's Theorem, which can be generalized to weakly compact cardinals.

Highlights limitations of extending combinatorial theorems to uncountable settings.

Abstract

We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.