Sum-free sets in finite groups

Question

Suppose $G$ is a group, $S \subset G$. Let’s call $S$ sum-free iff $\forall a, b \in S$ we have $ab \notin S$. Do there exist such $\epsilon > 0$, such that every sufficiently large finite group $G$ has a sum-free subset of cardinality $\geq \epsilon|G|$?

It was proved by Erdos using a very neat probabilistic approach, that for cyclic $G$ it is sufficient to take $\epsilon = \frac{1}{9}$. However I do not know what happens here in the case when the group is non-cyclic.

Answer 1

The answer is no -- this was shown in Gowers' paper "Quasirandom Groups".

From the abstract:

Babai and Sós have asked whether there exists a constant $c>0$ such that every finite group $G$ has a product-free subset of size at least $c|G|$: that is, a subset $X$ that does not contain three elements $x$, $y$ and $z$ with $xy=z$. In this paper we show that the answer is no.

Specifically, it is shown that

for sufficiently large $q$, the group $\mathrm{PSL}_2(q)$ has no product-free subset of size $Cn^{8/9}$, where $n$ is the order of $\mathrm{PSL}_2(q)$.

(taken from page 2.)