Densities, submeasures and partitions of groups

Abstract

In 1995 in Kourovka notebook the second author asked the following problem: it is true that for each partition $G=A_1\cup \cdots \cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=F{A}_i{A}_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups.

14 pages (this is an update of the preceding version)

References in corpus (4)

• Completeness of translation-invariant ideals in groups
• A Note on Combinatorial Derivation
• Extremal densities and measures on groups and $G$-spaces and their combinatorial applications
• Combinatorial Derivation