The number of additive triples in subsets of abelian groups

Abstract

AbstractA set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elementsx,y,zwithx+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 setAof elements of an abelian groupGhas cardinalitya. How many Schur triples mustAcontain? Moreover, which sets ofaelements ofGhave the smallest number of Schur triples? In this paper, we answer these questions for various groupsGand ranges ofa.