OPEN
Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$, so that there are no solutions to
\[a+b=c\]
with $a,b,c\in B$. Estimate $f(n)$.
Erdős gave a simple proof that shows $f(n) \geq n/3$. Alon and Kleitman
[AlKl90] improved this to $f(n)\geq \frac{n+1}{3}$, and Bourgain
[Bo97] further improved this to $\frac{n+2}{3}$. The best lower bound known is
\[f(n)\geq \frac{n}{3}+c\log\log n\]
for some constant $c>0$, due to Bedert
[Be25b]. The best upper bound known is
\[f(n) \leq \frac{n}{3}+o(n),\]
due to Eberhard, Green, and Manners
[EGM14].
