---

Claim. For every $\varepsilon>0$ and all large $n$ there is a set of $n$ positive integers in which every subset of more than $(\frac13+\varepsilon)n$ elements contains $x,y,z$ with $x+y=z$ and $x\ne y$ (Theorem 1.1, p. 2 of arXiv v3, with the stronger property stated in its introduction). In the notation of Problem 792,

$$f(n)\le\frac n3+o(n),$$

in the distinct-summand form, so the bound holds under both conventions for sum-free sets. With the lower bound $n/3-O(1)$ for every set of $n$ integers (Erdős's Theorem 2 applied to the nonzero elements), and Bourgain's $(n+2)/3$ for sets of positive integers, the main term is $f(n)=n/3+o(n)$ and $f(n)/n\to\frac13$; the paper notes that $f$ is subadditive, $f(m+n)\le f(m)+f(n)$ by the set $A\cup MB$ for large $M$, so one set with no sum-free subset larger than $(\frac13+\varepsilon)|A|$ suffices. The distinct-summand form also answers Erdős's question of 1965 whether excluding $x=y$ allows $f(n)=[(n+2)/2]$: it does not. The proof reduces to a local problem for a weight function on $\mathbb Z/Q\mathbb Z\times[0,1]$ and uses the arithmetic regularity lemma; it is not checked in this corpus. S. Eberhard, B. Green and F. Manners, Sets of integers with no large sum-free subset, Ann. of Math. (2) 180 (2014), no. 2, 621--652, arXiv:1301.4579 (v1 19 January 2013; v3 29 July 2026), cited as [EGM14] on the problem page. Library home eberhard_2014_sets_integers_no_large_sum_free; result page Theorem 1.1. The earlier constants $\sigma\le7/15$, $3/7$, $12/29$, $2/5$, $11/28$ and $11/28-\varepsilon$ are listed on the problem page.

Covers. The upper bound $f(n)\le n/3+o(n)$. Erdős's lower bound (a pending claim) fixes the main term $f(n)=n/3+o(n)$ with it, and the accepted Bourgain's Proposition 1.3 does so on sets of positive integers only. Not covered: the second-order term $f(n)-n/3$, for which the best lower bound is $c\log\log n$ (Bedert's preprint, claimed) and no upper bound sharper than $o(n)$ is in hand.

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

Acceptance. Refereed: the paper is the publisher's version of record in the Annals of Mathematics (Crossref); this page is named by the first arXiv posting, 19 January 2013; the statement is that of the 2026 arXiv revision v3, which is not compared with the journal text. The site's curator, Thomas F. Bloom, credits the best upper bound to Eberhard, Green and Manners in the problem page's commentary (label OPEN, page last edited 23 January 2026), and Bedert's preprint (p. 2) restates it as the best upper bound; the problem is not marked settled there and a citation is not a review, so neither credit is listed as reviewed. The statement is checked; the proof is not checked in this corpus.