# 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…

## 4 Citations

### Asymptotic formula for 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 there is no solution of the equation x+ y = z, with x, y, z belonging to the set A. Let SF (G) the sel of all sum-free…

### Counting sum-free sets in abelian groups

- Mathematics
- 2012

In this paper we study sum-free sets of order m in finite abelian groups. We prove a general theorem about independent sets in 3-uniform hypergraphs, which allows us to deduce structural results in…

### Random sum-free subsets of abelian groups

- Mathematics
- 2011

We characterize the structure of maximum-size sum-free subsets of a random subset of an abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each…

### 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…

### 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…

### Maximum number of sum-free colorings in finite abelian groups

- MathematicsIsrael Journal of Mathematics
- 2018

An r-coloring of a subset A of a finite abelian group G is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements a, b, c ∈ A with a + b = c. We investigate…

### 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…

### On some problems in Additive number theory

- Mathematics
- 2005

In this thesis we discuss some problems relating the properties of a set A and those of A + A, when A is a subset of an abelian group. Given a finite abelian group G and A ⊂ G, we say A is 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…

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