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13.45 (1995)

Solved

Every infinite group $G$ of regular cardinality $\mathfrak{m}$ can be partitioned into two subsets $G = A_1 \cup A_2$ so that $A_1 F \neq G$ and $A_2 F \neq G$ for every subset $F \subset G$ of cardinality less than $\mathfrak{m}$. Is this statement true for groups of singular cardinality?

Contributor: I. V. Protasov

Progress

No, not always. The answer depends on the algebraic structure of $G$. In particular, this is true for a free group, but the statement does not hold for every Abelian group $G$ of singular cardinality (I. Protasov, S. Slobodianiuk, Quest. Answers Gen. Topology, 33, no. 2 (2015), 61–70).

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