The covering number of the difference sets in partitions of \(G\)-spaces and groups (Q904676)

In this paper, the authors give two partial solutions to a problem of Protasov (see [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 13th augm. ed. Novosibirsk: Institute of Mathematics. Russian Academy of Sciences, Siberian Division (1995; Zbl 0838.20001)]). Specifically, it is shown that if a group \(G\) is partitioned into \(n\) subsets \(A_1\) through \(A_n\), then there is a subset \(F\) of \(G\) with \(|F| \leq n\) so that \(G = FA_iA^{-1}_{i}A_i\). The authors begin with a discussion of the original problem posed by Protasov, and of a natural generalization of that problem to \(G\)-spaces. They then state and prove the result stated above, as well as an analoguous result for \(G\)-spaces. The main result of the paper is actually a somewhat stronger statement applicable only to the group case, and it is proven separately from the more general theorem.