Problem Statement
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\[f(n)=2^{(1+o(1))\frac{n}{2}}?\]
Categories:
Number Theory
Progress
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.
Source: erdosproblems.com/748 | Last verified: January 16, 2026