  Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in
\mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset $F$ of $S$ such
that $S=F ^{-1} \{x \in S: x A \bigcap A\neq\emptyset\}$ and $|F |\leq n$?
  We give an affirmative answer provided that either $S$ is finite or $n=2$.
