Hindman-like theorems with uncountably many colours and finite monochromatic sets

TL;DR

This paper extends Hindman-like theorems to uncountably many colors, demonstrating that finite monochromatic sumsets can be guaranteed in large Boolean groups, generalizing and strengthening previous results.

Contribution

It generalizes Komjáth's result on monochromatic finite sumsets in uncountable partitions, establishing the strongest possible form of such theorems.

Findings

Monochromatic finite sumsets exist in large Boolean groups with uncountably many colors.

The generalization of Komjáth's result is proven to be optimal.

Finite monochromatic sets can be guaranteed under broader conditions.

Abstract

A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums $\{x_1+\cdots+x_n\big|n\in\mathbb N\wedge x_1,\ldots,x_n\in X\text{ are distinct}\}$ is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) $X$. On the other hand, a recent result of Komj\'ath states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form $\mathrm{FS}(X)$, for $X$ of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komj\'ath's result, and we show that, in a sense, this generalization is the strongest possible.