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

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.