PROVED
This has been solved in the affirmative.
Let $f(n)$ count the number of sum-free $A\subseteq \{1,\ldots,n\}$, i.e. $A$ contains no solutions to $a=b+c$ with $a,b,c\in A$. Is it true that\[f(n)=2^{(1+o(1))\frac{n}{2}}?\]
The
Cameron-Erdős conjecture. It is trivial to see that $f(n) \geq 2^{\frac{n}{2}}$, considering all subsets of $[n/2,n]$.
This is true, and in fact $f(n) \ll 2^{n/2}$, which was proved independently by Green
[Gr04] and Sapozhenko
[Sa03]. In fact, both papers prove the stronger asymptotic $f(n) \sim c_n 2^{n/2}$, where $c_n$ takes on one of two values depending on the parity of $n$.
See
[877] for the maximal case.
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T. F. Bloom, Erdős Problem #748, https://www.erdosproblems.com/748, accessed 2026-01-16
