Problem 792

Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\rvert \geq f(n)$ , so that there are no solutions to

a+b=c

with $a,b,c\in B$ . Estimate $f(n)$ .

Source: erdosproblems.com/792

Additive combinatorics

Wiki page

Status

Open

No claim settles this problem.

Open. The main term is determined: $f(n)\ge n/3$ by Erdős's Theorem 2 of 1965 ( $n-1$ in place of $n$ when $0\in A$ ; a proceedings volume with no refereeing evidence, so a pending partial claim on its claim page (Erdős, 1965)), the bound $(n+2)/3$ for sets of $n\ge3$ positive integers by Proposition 1.3 of Bourgain (Israel J. Math. 97 (1997), refereed; not held; stated for any set of positive integers and proved for $n\ge3$ ; an accepted partial claim on its claim page (Bourgain, 1997)) and $f(n)\le n/3+o(n)$ by Theorem 1.1 of Eberhard, Green and Manners (Ann. of Math. (2) 180 (2014), refereed; an accepted partial claim on its claim page (Eberhard, Green and Manners, 2013)). The second-order term is open: the best lower bound is $f(n)\ge n/3+c\log\log n$ , Theorem 1.2 of Bedert's 2025 preprint (arXiv:2502.08624v1; the site's commentary adopts it; a pending partial claim on its claim page (Bedert, 2025)), after Bourgain's $(n+2)/3$ on sets of positive integers and the $(n+1)/3$ of Alon and Kleitman on sets of nonzero integers (1990; a chapter in a tribute volume with no refereeing evidence, so a pending partial claim on its claim page (Alon and Kleitman, 1990)), and no upper bound sharper than $o(n)$ is in hand. A proof claim on the site's tab (9 September 2026), to which the site gives no kind, by Bedert, worked out with GPT 6 Astra as the tab names it, asserts $n/3+(\log n)^{1/3-o(1)}$ and is recorded as a pending partial claim on its claim page (Bedert, 2026). The site's label was OPEN on 2026-09-18, and none of the claim pages is a full claim. "Estimate $f(n)$ " is a request with no truth value; read on this page, the label concerns an estimate whose main term is settled and whose second-order term is the open question, the reading the site's own commentary takes.

Er65: Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math. VIII (Theory of Numbers), Amer. Math. Soc. (1965), 181--189, DOI 10.1090/pspum/008/0174539; Theorem 2 with condition (27), printed pp. 186--187; the Klarner example and the two conventions, p. 187; the Additions, p. 190. Library home: Erdos 1965; result page Theorem 2.
Er73: Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), North-Holland (1973), 117--138; Section 9, printed p. 129. Library home: Erdos 1973; result page Section 9.
Er92c: Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34--50; printed pp. 46--47, display (31) and the Alon--Kleitman sentence. Library home: Erdős 1992.
Va99: Various, Some of Paul's favorite problems. Booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 1.22 a). Library home: Various 1999; result page Problem 1.22.
AlKl90: Alon, N. and Kleitman, D. J., Sum-free subsets. In: A Tribute to Paul Erdős (A. Baker, B. Bollobás and A. Hajnal, eds.), Cambridge Univ. Press (1990), 13--26, DOI 10.1017/CBO9780511983917.003 (Crossref record read). Proposition 1.1, pp. 13--14; the $12/29$ remark, p. 14. A scan of the chapter is posted on the first author's publication page. Library home: Alon and Kleitman 1990; result page Proposition 1.1.
Bo97: Bourgain, J., Estimates related to sumfree subsets of sets of integers. Israel J. Math. 97 (1997), no. 1, 71--92, DOI 10.1007/BF02774027 (Crossref record read). Proposition 1.3, printed p. 72; its proof, pp. 74--76, with the conclusion (3.24) for $|B|\ge3$ on p. 76. Not held. Library home: Bourgain 1997; result page Proposition 1.3.
EGM14: Eberhard, S., Green, B. and Manners, F., Sets of integers with no large sum-free subset. Ann. of Math. (2) 180 (2014), no. 2, 621--652, DOI 10.4007/annals.2014.180.2.5; arXiv:1301.4579 (v1 19 January 2013; v3 29 July 2026, whose comment says it "corrects a very small inaccuracy in Lemma 6.3"). Theorem 1.1, p. 2 of v3. Library home: Eberhard, Green and Manners 2014; result page Theorem 1.1.
Be25b: Bedert, B., Large sum-free subsets of sets of integers via $L^1$ -estimates for trigonometric series. arXiv:2502.08624v1 (12 February 2025; 37 pages). Problem 1.1 and Theorem 1.2, p. 2; Theorem 2.2, p. 3. Library home: Bedert 2025; result page Theorem 1.2.
FGY26: Franchi, L., Gowers, W. T. and Yip, F., Product-free subsets of $(0,1)$ . arXiv:2607.06073v1 (7 July 2026; 53 pages). The continuous analog named in the thread, known here from its abstract; not held.

Problem 792

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Proof claims

Discussion