The previous results was not much better than |A|/3. The current, just proved, result shows that the largest subset is |A|/3 + c log(log(|A|)).
For example, the set {1,2,3} is not sum-free (1+2 = 3) but the subset {2,3} is sum-free (2+3 \notin {2,3}).
Yes, it seems to me we are focusing mainly about sets, not addition. Addition is secondary. Mainly I'm debating the title. The word "set" ought to be in the title too. I guess not a big deal.
So this problem is really more about addition than about sets, as the mathematicians who worked on it will say: the amount of set theory it involves is very little/almost nonexistent, while the properties of addition it involves are fairly deep.
(But sure, no harm if sets were mentioned in the title, I guess!)