In this talk we discuss completely syndetic (CS) sets in discrete groups - subsets that for every 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 integers, and as a result characterize subsets of integers whose closure in the Stone-Cech compacitifcation of Z contains the smallest two sided ideal. Finally, we show that CS sets can have an arbitrarily small density.

This talk is based upon arXiv:2506.18784, a joint work with Guy Salomon and Ariel Yadin.

TBD