SOLVED
Let $f_m(n)$ count the number of maximal sum-free subsets $A\subseteq\{1,\ldots,n\}$ - that is, there are no solutions to $a=b+c$ in $A$ and $A$ is maximal with this property. Estimate $f(n)$ - is it true that $f_m(n)=o(2^{n/2})$?
A problem of Cameron and Erdős, who proved that $f_m(n)>2^{n/4}$, and also asked whether
\[f_m(n)=o(f(n)),\]
where $f(n)$ counts the number of all (not necessarily maximal) sum-free sets. Luczak and Schoen
[LuSc01] proved that there exists a constant $c<1/2$ such that
\[f_m(n)<2^{cn},\]
resolving these questions. Balogh, Liu, Sharifzadeh, and Treglown
[BLST14] proved that
\[f_m(n)=2^{(\frac{1}{4}+o(1))n},\]
which the same authors
[BLST18] later improved to
\[f_m(n)=(C_n+o(1))2^{n/4},\]
where $C_n$ is some explicit constant depending only on $n\pmod{4}$.
See [748] for the non-maximal case.
