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Source. Theorem 2, p. 2, of Ben Green, The Cameron-Erdős conjecture, Bull. London Math. Soc. 36 (2004), no. 6, 769--778, cited from the arXiv manuscript math/0304058v1 (4 April 2003) whose pages the labels below follow, as identified on the source card.

Statement

A set $A$ of integers is sum-free when there are no $x,y,z\in A$ with $x+y=z$ ($x=y$ allowed). Write $[N]=\{1,\ldots,N\}$ and $\mathrm{SF}(N)$ for the collection of sum-free subsets of $[N]$ (p. 1).

Theorem 2 (p. 2). "The number of sum-free subsets of $[N]$ is asymptotically $c(N)2^{N/2}$, where $c(N)$ takes two different constant values according as $N$ is odd or even." (quoted)

That is, there are constants $c_{\mathrm{odd}}$ and $c_{\mathrm{even}}$, with $c_{\mathrm{odd}}\ne c_{\mathrm{even}}$, such that $|\mathrm{SF}(N)|/(c(N)2^{N/2})\to1$ as $N\to\infty$, where $c(N)$ is $c_{\mathrm{odd}}$ for odd $N$ and $c_{\mathrm{even}}$ for even $N$. The paper does not give the values of the two constants. In particular $|\mathrm{SF}(N)|=O(2^{N/2})$, which is Conjecture 1 of Cameron and Erdős (p. 1), the form stated in the abstract.

Proof pointer

Section 2 (p. 2) outlines the strategy and Sections 3 and 4 (pp. 2--11) carry it out. Proposition 6 (p. 7) covers every sum-free subset of $[N]$ by one of $2^{o(N)}$ sets with $o(N^2)$ additive triples; a structure theorem for large sets with few additive triples (Proposition 7, p. 8) then gives Corollary 13 (p. 10): all but $o(2^{N/2})$ sum-free subsets of $[N]$ consist of odd numbers or lie in $\{\lceil(N+1)/3\rceil,\ldots,N\}$. The final step (p. 11) is not proved in this paper: it combines Corollary 13 with the count, due to Cameron and Erdős (their 1990 paper, the paper's reference [4]), of the sum-free subsets of $\{\lceil(N+1)/3\rceil,\ldots,N\}$ as asymptotically $c(N)2^{N/2}$. On the way, Proposition 12 (p. 10) rederives from Propositions 6 and 7 the earlier bound $|\mathrm{SF}(N)|=2^{N/2+o(N)}$ of Alon, Calkin, and Erdős and Granville (display (1), p. 1).

Dependencies

Proposition 6, Corollary 13 and the Cameron-Erdős count of sum-free subsets of the top interval, which the paper cites and does not prove. Read depth: claims checked; the statement was read on p. 2 and the closing argument on p. 11; the proofs of Sections 3 and 4 were read for their structure only.

Bears on

• Problem 748: the theorem gives $f(n)\sim c(n)2^{n/2}$ for the problem's count $f(n)$ of sum-free subsets of $\{1,\ldots,n\}$, the bound $f(n)=O(2^{n/2})$ named on the problem page as the Cameron-Erdős conjecture, and with the lower bound $f(n)\ge2^{\lceil n/2\rceil}$ from the subsets of $(n/2,n]$ the asked exponent form $f(n)=2^{(1+o(1))n/2}$. The exponent form alone is the older bound the paper attributes to Alon, Calkin, and Erdős and Granville.
• Problem 877: background only. The theorem counts all sum-free subsets, not the maximal ones the problem counts; it shows that the problem's question $f_m(n)=o(2^{n/2})$ is the same as asking that maximal sum-free subsets be a vanishing proportion of all sum-free subsets, and it gives no bound on $f_m(n)$ beyond the trivial $f_m(n)\le f(n)$.