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

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 ⊆ G such that the set of its finite sums {x₁ + ⋯ + x_n | n ∈ ℕ Λ x₁,⋯, x_n ∈ X 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 Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form FS(X), for X of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth's result, and we show that, in a sense, this generalization is the strongest possible.