On the largest sum-free subset problem in the integers

Let $A\subset {\mathbb{Z}}_{>0}$ of size $n$. It is conjectured that for any $C>0$ and $n$ large enough that $A$ contains a sum-free subset of size at least $n\mathrm{/}3+C$. We study this problem and find an alternate proof of Bourgain’s result that one make take $C=2\mathrm{/}3$.