The number of additive triples in subsets of abelian groups

Wojciech Samotij, Benny Sudakov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)495-512
Number of pages18
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number3
StatePublished - 1 May 2016

All Science Journal Classification (ASJC) codes

  • General Mathematics


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