Partitions of groups into sparse subsets. (English) Zbl 1258.20036

Protasov, Igor

Algebra Discrete Math. 13, No. 1, 107-110 (2012).

Summary: A subset \(A\) of a group \(G\) is called sparse if, for every infinite subset \(X\) of \(G\), there exists a finite subset \(F\subset X\), such that \(\bigcap_{x\in F}xA\) is finite. We denote by \(\eta(G)\) the minimal cardinal such that \(G\) can be partitioned in \(\eta(G)\) sparse subsets. If \(|G|>(\kappa^+)^{\aleph_0}\) then \(\eta(G)>\kappa\), if \(|G|\leq\kappa^+\) then \(\eta(G)\leq\kappa\). We show also that \(\text{cov}(A)\geq\text{cf}|G|\) for each sparse subset \(A\) of an infinite group \(G\), where \(\text{cov}(A)=\min\{|X|:G=XA\}\).

MSC:

20F99	Special aspects of infinite or finite groups
03E75	Applications of set theory
03E50	Continuum hypothesis and Martin’s axiom
20F05	Generators, relations, and presentations of groups
05D10	Ramsey theory

Keywords:

infinite groups; sparse subsets of groups; partitions of groups