## Counting sum-free sets in abelian groups.(English)Zbl 1332.11030

In this paper, sum-free sets of order $$m$$ in finite abelian groups are studied. Sum-free subsets of abelian groups are central objects of interest in Additive Combinatorics, and have been studied intensively in recent years. The main questions are as follows: What are the largest sum-free subsets of a group? How many sum-free sets are there? And what does a typical such set look like?
The main results obtained in this paper are the following:
Theorem 1.1. For every prime $$q\equiv 2\pmod 3$$, there exists a constant $$C(q)>0$$ such that the following holds. Let $$G$$ be an abelian group of Type I($$q$$) and order $$n$$, and let $$m\geq C(q)\sqrt{n\log n}$$. Then almost every sum-free subset of $$G$$ of size $$m$$ is contained in a maximum-size sum-free subset of $$G$$, and hence $|SF(G,m)|=\lambda _{q}(\#\{\text{elements\, of\, $$G$$\, of\, order\, $$q$$}\}+o(1))\left( \begin{matrix} \mu (G)n \\ m \end{matrix} \right)\quad\text{as }n\to \infty,$ where $$\lambda _{q}=1$$ if $$q=2$$ and $$\lambda _{q}=1/2$$ otherwise.
and
{Theorem 1.3} For every $$\varepsilon >0$$, there exists a constant $$C=C(\varepsilon )$$ such that the following holds. If $$\mathcal{G}$$ is an $$n$$ -vertex $$d$$-regular graph with $$\lambda (\mathcal{G})\geq -\lambda$$, then $I(G,m)\leq \left( \begin{matrix} (\frac{\lambda }{d+\lambda }+\varepsilon )n \\ m \end{matrix} \right)$ for every $$m\geq Cn/d$$.
There is a very elaborate description of the historical context of such problems (including a large Bibliography).

### MSC:

 11B75 Other combinatorial number theory 11B13 Additive bases, including sumsets 20K01 Finite abelian groups 05D10 Ramsey theory
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