Recent progress in subset combinatorics of groups

Abstract

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra ${\mathcal{P}}_G$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $\beta G$ of $G$ , the combinatorial derivation.

Large, small, thin, thick, sparse and scattered subsets of groups; descriptive complexity; Boolean algebra of subsets of a group; Stone-$\check{C}$ech compactification; ultracompanion; Ramsey-product subset of a group; recurrence; combinatorial derivation. arXiv admin note: text overlap with arXiv:1704.02494