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Claim. Every set $B$ of positive integers with $|B|\ge3$ has a sum-free subset of at least $\frac13(|B|+2)$ elements, sum-free in the convention of Problem 792: the paper calls $A$ sumfree when $(A+A)\cap A=\emptyset$ (printed p. 71), so $a+b=c$ is forbidden with $a=b$ included. This bounds the site's $f(n)$ only on sets of positive integers: with $0$ added it gives $(n+1)/3$ when the other elements are positive, sets with negative elements are not treated, and $\{0,1,2\}$, whose largest sum-free subset has one element, shows $f(3)\le1<\frac53$. On sets of positive integers it improves Alon and Kleitman's $|A|>n/3$ and Erdős's $n/3$. The statement (Proposition 1.3, printed p. 72) carries no size restriction; the proof's conclusion (3.24) on p. 76 is for $|B|\ge3$, which the bound needs ($\{1,2\}$ has only singleton sum-free subsets, and $1<\frac43$), and $f(n)\ge(n+2)/3$ for $n\ge3$ is the form in which Eberhard, Green and Manners (2014, pp. 1--2) quote it; Bedert (2025, p. 2) states $S(N)\ge(N+2)/3$ with no size condition. The proof (pp. 74--76) writes Erdős's rotation argument as the Fourier minorization $S(B)\ge\frac{|B|}3+\max_x\sum_{m\in B}(f-\frac13)(mx)$ for the indicator $f$ of $(1/3,2/3)$ and shows that the maximum exceeds $\frac13$ by a case analysis on the three smallest elements of $B$. The paper states the bound for positive integers; it transfers to a set containing $0$, with $n-1$ in place of $n$, only when the other elements are positive, and not to sets with negative elements. J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math. 97 (1997), no. 1, 71--92, cited as [Bo97] on the problem page. Library home bourgain_1997_estimates_related_sumfree_subsets_sets_integers; result page Proposition 1.3.

Covers. The lower bound $(n+2)/3$ for sets of $n\ge3$ positive integers. Not covered: sets with $0$ or negative elements, on which the site's $f(n)$ is also taken; the second-order term, for which Bedert's preprint (claimed) asserts $c\log\log n$; and the upper bound.

Depends on. No page of this wiki; the proof is self-contained.

Acceptance. Refereed: the paper is the publisher's version of record in the Israel Journal of Mathematics (Crossref: issue dated December 1997, with no day, so this page is named by the month). The site's curator, Thomas F. Bloom, credits the improvement to $(n+2)/3$ to Bourgain in the problem page's commentary (label OPEN, page last edited 23 January 2026); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The statement and the conclusion (3.24) are checked and the proof is followed for its structure; the numerical case bounds (3.9)--(3.23) are not recomputed in this corpus.