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The number of additive triples in subsets of abelian groups

Published online by Cambridge University Press:  26 January 2016

WOJCIECH SAMOTIJ  and
BENNY SUDAKOV
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel. e-mail: samotij@post.tau.ac.il
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland. e-mail: benjamin.sudakov@math.ethz.ch
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 3 , May 2016 , pp. 495 - 512
Copyright
Copyright © Cambridge Philosophical Society 2016 

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