# Sum-free sets in abelian groups

@article{Green2003SumfreeSI, title={Sum-free sets in abelian groups}, author={Ben Green and Imre Z. Ruzsa}, journal={Israel Journal of Mathematics}, year={2003}, volume={147}, pages={157-188} }

LetA be a subset of an abelian groupG with |G|=n. We say thatA is sum-free if there do not existx, y, z εA withx+y=z. We determine, for anyG, the maximal densityμ(G) of a sum-free subset ofG. This was previously known only for certainG. We prove that the number of sum-free subsets ofG is 2(μ(G)+o(1))n, which is tight up to theo-term. For certain groups, those with a small prime factor of the form 3k+2, we are able to give an asymptotic formula for the number of sum-free subsets ofG. This…

## 13 Citations

Sum-free set in finite abelian groups

- Mathematics
- 2005

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise…

The number of additive triples in subsets of abelian groups

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

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…

Large Sum-free Sets in Abelian Groups

- Mathematics
- 2005

Let A be a subset of a finite abelian group G. We say that A is sum-free if the equation x + y = z, has no solution (x, y, z) with x, y, z belonging to the set A. In this paper we shall characterise…

ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

- Mathematics
- 2008

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We…

On the structure of large sum-free sets of integers

- MathematicsIsrael Journal of Mathematics
- 2018

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x + y = z. In this paper, we provide a structural characterisation of sum-free…

N ov 2 00 7 Product-free subsets of groups , then and now

- Mathematics
- 2018

Let G be a group. A subset S of G is product-free if there do not exist a, b, c ∈ S (not necessarily distinct) such that ab = c. One can ask about the existence of large product-free subsets for…

A ug 2 00 7 Product-free subsets of groups , then and now

- Mathematics
- 2008

Let G be a group. A subset S of G is product-free if there do not exist a, b, c ∈ S (not necessarily distinct) such that ab = c. One can ask about the existence of large product-free subsets for…

On Maximal Sum-Free Sets in Abelian Groups

- MathematicsElectron. J. Comb.
- 2022

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby…

A pr 2 00 8 Generalizations of product-free subsets

- Mathematics
- 2008

In this paper, we present some generalizations of Gowers’s result about product-free subsets of groups. For any group G of order n, a subset A of G is said to be product-free if there is no solution…

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