A particular case of the Hindman–Galvin–Glazer theorem states that, for every partition of an infinite abelian group
G
G
into two cells, there will be an infinite
X
⊆
G
X\subseteq G
such that the set of its finite sums
{
x
1
+
⋯
+
x
n
∣
n
∈
N
∧
x
1
,
…
,
x
n
∈
X
are distinct
}
\{x_1+\cdots +x_n \mid 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
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
F
S
(
X
)
\mathrm {FS}(X)
, for
X
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.