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arXiv · 1507.03764

The number of additive triples in subsets of abelian groups

Wojciech Samotij·Benny Sudakov
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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 $a$.

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Wojciech Samotij, Benny Sudakov. 2016-07-20. The number of additive triples in subsets of abelian groups. https://doi.org/10.1017/s0305004115000821

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