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Claim. There is a constant $c>0$ such that every finite set $A\subset\mathbb Z$ has a sum-free subset of at least $\frac{|A|}3+c\log\log|A|$ elements (Theorem 1.2, p. 2), sum-free in the convention of Problem 792, with $a=b$ allowed. In the problem's notation, for all large $n$,

$$f(n)\ge\frac n3+c\log\log n,$$

which answers the paper's Problem 1.1, Problem 1 on Green's list of open problems: there is a function $\omega(n)\to\infty$ with $f(n)\ge n/3+\omega(n)$. It improves [[problems/additive_combinatorics/E0792/claims/1997_12_01_bourgain|Bourgain's $(n+2)/3$]] and [[problems/additive_combinatorics/E0792/claims/1990_01_01_alon_kleitman|Alon and Kleitman's $(n+1)/3$]]. The theorem is deduced from Theorem 2.2 (p. 3), a Freiman-isomorphic copy $B$ of $A$ with

$$\max_x\sum_{b\in B}\bigl(\varphi-\tfrac13\bigr)(bx)\gg\log\log|B|$$

for the indicator $\varphi$ of $(1/3,2/3)$, through Bourgain's Fourier expansion of Erdős's rotation argument, inverse theorems for sets whose Fourier transform has small $L^1$ norm, a dense model and the distribution of $A$ modulo small primes; the proof (Sections 4--9) is not checked in this corpus. B. Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series, arXiv:2502.08624v1 (12 February 2025; 37 pages), cited as [Be25b] on the problem page. Library home bedert_2025_large_sum_free_subsets_sets_integers; result page Theorem 1.2.

Covers. A lower bound for the second-order term, $f(n)-n/3\gg\log\log n$. Not covered: the upper bound, which is Eberhard, Green and Manners's $o(n)$, and the true order of the second-order term; the author's later claim of 2026 asserts the larger $(\log n)^{1/3-o(1)}$.

Depends on. No page of this wiki; the argument rests on the literature the paper cites.

Standing. Claimed. The paper is a preprint, the only arXiv version on 2026-09-18, with no journal record (Crossref bibliographic query of that date) and no published independent review or dispute; three later preprints cite it, none a review. The site's curator, Thomas F. Bloom, calls it the best lower bound known in the problem page's commentary (label OPEN, page last edited 23 January 2026), which adopts the bound without recording an examination of the proof; without a refereed version or a reviewer independent of the author on record, the page stays claimed. The statement is checked; the proof is not checked in this corpus.