Completely Syndetic Sets in Discrete Groups

We study completely syndetic (CS) sets in discrete groups - subsets that for every natural n admit finitely many left translates that jointly cover every n-tuple of group elements. While for finitely-generated groups, the non-virtually nilpotent ones admit a partition into two CS sets, we show that virtually abelian groups do not. We also characterize CS subsets of the group of integers Z, and as a result characterize subsets of Z whose closure in the Stone-Cech compactification contains the smallest two sided ideal. Finally, we show that CS sets can have an arbitrarily small density.