The number of additive triples in subsets of abelian groups

TL;DR

This paper investigates the minimum number of Schur triples in subsets of finite abelian groups of a given size, extending the understanding of sum-free sets to more general configurations and group structures.

Contribution

It provides new results on the minimal number of Schur triples in subsets of abelian groups for various sizes and groups, generalizing previous sum-free set studies.

Findings

Determines the minimal number of Schur triples for different group sizes and types.

Identifies the structure of subsets with the fewest Schur triples.

Extends classical results on sum-free sets to broader configurations.

Abstract

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we answer these questions for various groups $G$ and ranges of $a$.