Counting configuration-free sets in groups

TL;DR

This paper develops new methods for asymptotically counting subsets in groups that avoid specific configurations, including solutions to equations and linear patterns, using hypergraph containers and arithmetic removal lemmas.

Contribution

It introduces novel counting techniques for configuration-free sets in both abelian and non-abelian groups, expanding the scope of combinatorial group theory.

Findings

Asymptotic formulas for configuration-free subsets in groups

Extension of hypergraph container methods to group settings

Results on random subsets avoiding certain configurations

Abstract

We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.