Problem 792

../

claims/: The 6 claim pages of Problem 792, one per claimant's result; the problem's standing derives from them.

---

Statement. 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 $f(n)$.

Formulation. The site's wording as of 2026-09-18T15:09Z (page last edited 23 January 2026). The relation $a+b=c$ is forbidden for all $a,b,c\in B$, the case $a=b$ included, so no element of $B$ is twice another. This is the convention of Erdős's condition (27) of 1965, whose indices satisfy $1\le j_1\le j_2<j_3\le k$ (printed p. 186), of Alon and Kleitman ("no (not necessarily distinct) $a,b,c\in A$ such that $a+b=c$", p. 13) and of Bedert; Eberhard, Green and Manners state their upper bound in the stronger form that forbids only $x+y=z$ with $x\ne y$, which covers both conventions. The site's $A\subset\mathbb Z$ admits $0$, which lies in no sum-free set since $0+0=0$; for a set containing $0$ the question is the same question for its $n-1$ nonzero elements. The sources state the exact bounds $(n+1)/3$ for sets of $n$ nonzero integers and $(n+2)/3$ for sets of $n$ positive integers, and with $0$ allowed they hold with $n-1$ in place of $n$; the asymptotic statements are unaffected (an observation made here). Erdős posed the question for $n$ real numbers different from $0$; integer sets are real sets and his rotation proof works for reals, so the lower bound $n/3$ and the integer upper constructions below hold in either formulation, while the exact bounds $(n+1)/3$ and $(n+2)/3$ are stated in their sources for integers. The site's source keys are [Er65], [Er73], [Er92c] and [Va99, 1.22].

Status. 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), 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) 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). 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), 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), 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. 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.

Source. erdosproblems.com/792, accessed 2026-09-18T15:09Z: the problem page (OPEN, with the site's note that no finite computation can resolve it; last edited 23 January 2026; source keys [Er65], [Er73], [Er92c], [Va99, 1.22]; commentary citing [AlKl90], [Bo97], [Be25b], [EGM14] and Green's open problems list; indicators "Formalised statement? No" and "OEIS: Possible"), its three-comment discussion thread (25 and 26 July 2026) and its proof-claim tab with one proof claim (9 September 2026). Cite as: T. F. Bloom, Erdős Problem #792, https://www.erdosproblems.com/792, accessed 2026-09-18.

References.

• [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_extremal_problems_number_theory; 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_problems_results_combinatorial_number_theory; 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: erdos_1992_my_forgotten_problems_number_theory.
• [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_some_pauls_favorite_problems; 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_1990_sum_free_subsets; 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_estimates_related_sumfree_subsets_sets_integers; 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_2014_sets_integers_no_large_sum_free; 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_large_sum_free_subsets_sets_integers; 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.

Formalization. None in formal-conjectures: no file ErdosProblems/792.lean exists in google-deepmind/formal-conjectures (main, on 2026-09-18 and on 2026-10-07), and the page's indicator read "Formalised statement? No (create one)" on 2026-09-18. The community database (teorth/erdosproblems, 2026-09-18T15:04Z; its copy of 2026-10-06 agrees) records the problem open (last update 31 August 2025), the statement not formalized and an OEIS entry marked "possible".

Current assessment

The question (site formulation of 2026-09-18T15:09Z). The statement above; OPEN, with the site's note that no finite computation can resolve it; last edited 23 January 2026. The commentary credits the simple proof of $f(n)\ge n/3$ to Erdős [Er65], the improvements to $(n+1)/3$ and $(n+2)/3$ to Alon and Kleitman [AlKl90] and Bourgain [Bo97], the best lower bound $f(n)\ge n/3+c\log\log n$ to Bedert [Be25b] and the best upper bound $f(n)\le n/3+o(n)$ to Eberhard, Green and Manners [EGM14], and records that Green's list of open problems opens with this one. The thread (three comments): on 25 July 2026 a commenter points to a continuous version of the problem, also on Green's list, whose recent resolution proved a similar upper bound, linking [FGY26]; a second commenter asks whether that is the product version of the real case; the first replies that taking logarithms turns it into an additive problem. The proof-claim tab holds one proof claim (below). The community database record says open.

The origins. [Er65], printed p. 186, introduces the question as one of several that arose from attempts to improve the paper's Theorem 1, and defines the function in Erdős's words: "Let $a_1,a_2,\dots,a_n$ be $n$ real numbers all different from $0$. Denote by $f(n)$ the largest integer so that for every sequence $a_1,\dots,a_n$ one can always select $k=f(n)$ of them $a_{i_1},\dots,a_{i_k}$ so that (27) $a_{i_{j_1}}+a_{i_{j_2}}\ne a_{i_{j_3}}$, $1\le j_1\le j_2<j_3\le k$. THEOREM 2. $f(n)\ge n/3$." The proof (pp. 186--187) takes the set $I_r$ of $\alpha\in(0,T)$ with $a_r\alpha\pmod1$ between $1/3$ and $2/3$, whose measure is $T/3$ up to a bounded error, and finds an $\alpha$ at which at least $n/3$ of the $a_r\alpha\pmod1$ lie in $(1/3,2/3)$; "Clearly these $a$'s satisfy (27), which proves Theorem 2." Then (p. 187) the paper asks whether Theorem 2 can be improved: the sequence $1,2,\cdots,n$ shows $f(n)\le\lfloor(n+2)/2\rfloor$ in any case, and with $j_1=j_2$ permitted in (27) the bound "$f(n)\le\frac37n$" follows from the seven numbers $2,3,4,5,6,8,10$, no four of which avoid one being the difference of two others, multiplied by $10^r$, $r=1,\dots,k$. The construction is credited to D. Klarner, with an earlier, slightly weaker independent example by A. J. Hilton, and the paper guesses that excluding $j_1=j_2$ in (27) might give $f(n)=[(n+2)/2]$, remarking on the surprising difficulty of so simple a question. The page adds that Theorem 2 holds for any finite Abelian group and for measurable sets of reals, with $1/3$ best possible for measurable sets modulo $1$ and for residues modulo $p$. The Abelian-group remark is false as stated: Theorem 1.3 of [AlKl90] (p. 14) gives $s(B)>\frac27|B|$ for every set $B$ of nonzero elements of a finite Abelian group with the constant $2/7$ best possible, so no constant above $2/7$, and $1/3$ in particular, holds in every finite Abelian group (an observation made here from the two statements). [Er73], printed p. 129, restates the function with the same condition (9.1): "It is not hard to see that $f(n)\ge\frac13n$. This is almost certainly not best possible but Klarner and Hilton showed $f(n)<\frac12n$ even if we exclude $j_1=j_2$." [Er92c], printed p. 46: "A sequence of integers $b_1<b_2<\cdots$ is called sum free if the sum of two $b$'s never equals a third. In an old paper of mine I investigated the following question: Let $g(n)$ be the largest integer for which any sequence $a_1<a_2<\cdots<a_n$ contains a sum free subsequence of $g(n)$ terms. I proved [18] (31) $\frac n3\le g(n)\le\frac{3n}7$. Very recently Noga Alon and Kleitman improved (31), they proved $\frac n3<g(n)\le\frac{12}{29}n$", and p. 47: "The exact value of $g(n)$ is still not known and is I think an interesting question." [Va99], item 1.22 a): "Avoid $b_1+b_2=b_3$ with $b_i\in B$. The maximum of $k$ is somewhere between $n/3$ and $n/2$."

An observation made here on the two conventions, a finite check of the seven printed numbers and nothing more: among $\{2,3,4,5,6,8,10\}$ the largest subset with no solution of $a+b=c$ has three elements when $a=b$ is allowed (for instance $\{3,4,5\}$) and four when only distinct summands count ($\{2,3,4,8\}$, $\{3,4,5,10\}$ and others), so the bound $3n/7$ from Klarner's numbers needs the convention of the site's statement, as the 1965 sentence says, while the 1973 sentence's $f(n)<\frac12n$ "even if we exclude $j_1=j_2$" is not supported by the printed example, which gives only $4n/7$ in that convention (Hilton's example is not printed). Nothing is decided about what Klarner and Hilton showed; the upper bound now in force, [EGM14], holds in the stronger distinct-summand form.

The lower bounds. Theorem 2 of [Er65] gives $f(n)\ge n/3$ (the half-page rotation proof is not checked in this corpus); it is a pending partial claim on its claim page, since the Proceedings of Symposia in Pure Mathematics volume carries no refereeing evidence. Proposition 1.1 of [AlKl90], pp. 13--14: "Any set $B$ of $n$ non-zero integers contains a sum-free subset $A$ of cardinality $|A|>\frac13n$", the strict inequality being the improvement over Erdős, whose result the authors "learned later" had been "proved by Erdős more than twenty years ago"; since $|A|$ is an integer this is the site's $(n+1)/3$ (a chapter in a tribute volume with no refereeing evidence found, so a pending partial claim on its claim page; it covers sets with negative elements, which Bourgain's bound does not). The paper's Proposition 1.2 (p. 14) extends the bound to sequences of nonzero integers, its Theorem 1.3 gives $s(B)>\frac27|B|$ in every finite Abelian group with $2/7$ best possible, and p. 14 reports that the constant $1/3$ "cannot be replaced by $\frac{12}{29}$ (or any bigger constant)" and that the infimum of $s(B)/|B|$ over sequences is not attained. Proposition 1.3 of [Bo97], p. 72: "$S(B)\ge\frac13(|B|+2)$, for any $B\subset\mathbb Z_+$", where "$S(B)$ denotes the maximum size of a sumfree subset of $B$" and sumfree means $(A+A)\cap A=\emptyset$ (p. 71), the convention of the site's statement. 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 the maximum exceeds $\frac13$ by a case analysis on the three smallest elements of $B$, concluding (3.24) "for any $B\subset\mathbb Z_+$, $|B|\ge3$"; the statement's missing hypothesis $n\ge3$ is needed ($\{1,2\}$ has $S=1<\frac43$), and it is the form in which [EGM14], pp. 1--2, quotes it ("who showed $f(n)\geqslant\frac13(n+2)$ for $n\geqslant 3$ using an elaborate Fourier-analytic technique"), while [Be25b], p. 2, gives the bound with no size condition ("The best bound $S(N)\ge(N+2)/3$ before this work was established in a celebrated paper of Bourgain [4]"). The proof's numerical case bounds are not recomputed in this corpus. The bound is an accepted partial claim on its claim page, on the refereed publication. Theorem 1.2 of [Be25b], p. 2: "There exists some constant $c>0$ such that for all finite sets $A\subset\mathbb Z$ we have $S(A)\ge\frac{|A|}3+c\log\log|A|$. In particular, $S(N)\ge\frac N3+c\log\log N$", where $S(A)$ is the size of the largest sum-free subset of $A$ and $S(N)$ its minimum over sets of $N$ positive integers, the paper's form of $f(n)$. The paper states this as the answer to its Problem 1.1 ("Is there a function $\omega(N)\to\infty$ such that $S(N)\ge\frac N3+\omega(N)$?"), "listed as Problem 1 on Green's list [8] of 100 open problems"; the theorem is deduced from Theorem 2.2 (p. 3), a Freiman-isomorphic copy $B$ of $A$ with $\max_x\sum_{b\in B}(\varphi-\tfrac13)(bx)\gg\log\log|B|$ for the indicator $\varphi$ of $(1/3,2/3)$, through Bourgain's Fourier expansion, inverse theorems for sets with small $L^1$ norm of $\hat1_A$, a dense model and the distribution of $A$ modulo small primes (the paper's overview, pp. 3--5; the proof, Sections 4--9, is not checked in this corpus). Acceptance evidence for [Be25b]: the site's commentary calls it the best lower bound known; three 2025--2026 preprints cite it (Semantic Scholar), none a review. The bound is therefore recorded with the preprint qualification, as a pending partial claim on its claim page.

The upper bound. Theorem 1.1 of [EGM14], p. 2: "There is a set of $n$ positive integers with no sum-free subset of size greater than $\frac13n+o(n)$." The introduction explains that $f(m+n)\le f(m)+f(n)$ (the set $A\cup MB$ for large $M$), so $f(n)/n$ converges to $\sigma=\inf f(n)/n$ and one set with no sum-free subset larger than $(\frac13+\varepsilon)|A|$ suffices; the constructed set has the stronger property that "every subset $A'$ of size larger than $(\frac13+\varepsilon)|A|$ contains a solution to $x+y=z$ with $x\ne y$. This answers a further question asked in [Erd65]", the $[(n+2)/2]$ guess of p. 187. The proof (Sections 3--5 and Appendix A) 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. The published paper is Ann. of Math. (2) 180 (2014), 621--652 (Crossref), an accepted partial claim on its claim page; the statement quoted is that of the 2026 arXiv revision v3, which is not compared with the journal text. Before it, the constants $\sigma\le7/15$ (Hilton, printed "Hinton" in [EGM14] and [Be25b]), $3/7$ (Klarner), $12/29$ (Alon and Kleitman), $2/5$ (Malouf; also Füredi), $11/28$ (Lewko) and $11/28-\varepsilon$ (Alon) are listed on [EGM14], p. 2, and [Be25b], p. 2, with the trivial $f(n)\le\frac12(n+1)$.

The 2026 claim. The proof-claim tab carries one proof claim, to which the site gives no kind, submitted 2026-09-09 15:03:14 by Benjamin Bedert, the author of [Be25b], declaring the use of the model GPT 6 Astra, recorded on its claim page. It asserts $f(n)\ge n/3+(\log n)^{1/3-o(1)}$ by an approach the author had considered before the preprint's route and worked out in a conversation with the model, combining the preprint's dense model and partial sifting results with structural results for functions of small Fourier algebra norm in the manner of Sanders's papers; its note says the write-up is AI-generated and that a human-readable version is intended. The write-up is a document on a file-sharing service, not a citable source, and is not examined in this corpus; the two comments on the claim, a congratulation and a question about the write-up's contents, review nothing; the site's label and commentary do not adopt it. The thread's [FGY26] proves, per its abstract, that an open subset of $(0,1)$ with no $x,y,z$ with $xy=z$ has measure at most $1/3$, Green's Problem 3; it is the multiplicative analog of the real version of this problem and not the problem.

Search scope. None of the routes below found an upper bound sharper than $o(n)$, a refereed version of [Be25b], or a review or dispute of it.

• The site: problem page, discussion thread and proof-claim tab as read on 2026-09-18; the formal-conjectures directory listing (no file) and the community database, both on 2026-09-18.
• arXiv: the abstract pages of 1301.4579 (three versions; the journal DOI) and 2502.08624 (one version; no journal reference); the API queries abs:"sum-free subset" AND abs:integers sorted by date (16 records, titles read; the 2026 item is a counting problem, and nothing newer than [Be25b] concerns $f(n)$) and all:"Erdős Problem" AND (all:787 OR all:788 OR all:790 OR all:792) (no records); the abstract page of 2607.06073.
• Crossref: the records of [EGM14], [Bo97] and [AlKl90]; a bibliographic query for the title of [Be25b] (no journal record).
• Semantic Scholar: the citation lists of 2502.08624 (three records) and 1301.4579 (42 records; titles scanned; the two items on the problem itself, arXiv:2011.09963 (2020) and arXiv:2207.14210 (2022), predate [Be25b] and were not read).
• The first author's publication page for [AlKl90] (a scan of the chapter).
• The primary sources, at the pages cited: [Er65] pp. 186--187 and 190, [Er73] p. 129, [Er92c] pp. 46--47, [Va99] item 1.22, [AlKl90] pp. 13--14, [EGM14] pp. 1--3, [Be25b] pp. 1--5 and [Bo97] pp. 71--76.

Not searched: MathSciNet, zbMATH, Google Scholar, X.

Remaining gaps. (1) The second-order term is open between $c\log\log n$ and $o(n)$; the lower bound rests on an unrefereed preprint, whose journal version or independent review is the reopening condition for the qualification. (2) [Bo97] is not held; the proof of its Proposition 1.3 is checked for structure only, and the numerical case bounds (3.9)--(3.23) are not recomputed. (3) The tab's proof claim is not examined in this corpus and has no independent review; its claim page records it. (4) Proof coverage is claims checked throughout; no proof is reviewed, and the journal text of [EGM14] is not compared with the 2026 arXiv revision v3. (5) The determined main term does not close the estimate, whose second-order term is the open question.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• alon_1990_sum_free_subsets
• alon_1990_sum_free_subsets / construction_p15
• alon_1990_sum_free_subsets / proposition_1_1
• alon_1990_sum_free_subsets / proposition_1_2
• alon_1990_sum_free_subsets / proposition_4_1_prime
• alon_1990_sum_free_subsets / theorem_1_3
• bedert_2025_large_sum_free_subsets_sets_integers
• bedert_2025_large_sum_free_subsets_sets_integers / theorem_1_2
• bourgain_1997_estimates_related_sumfree_subsets_sets_integers
• bourgain_1997_estimates_related_sumfree_subsets_sets_integers / display_8_4
• bourgain_1997_estimates_related_sumfree_subsets_sets_integers / proposition_1_3
• bourgain_1997_estimates_related_sumfree_subsets_sets_integers / proposition_1_4
• bourgain_1997_estimates_related_sumfree_subsets_sets_integers / proposition_1_7
• eberhard_2014_sets_integers_no_large_sum_free
• eberhard_2014_sets_integers_no_large_sum_free / theorem_1_1
• erdos_1965_extremal_problems_number_theory
• erdos_1965_extremal_problems_number_theory / theorem_2
• erdos_1973_problems_results_combinatorial_number_theory
• erdos_1973_problems_results_combinatorial_number_theory / section_9
• erdos_1992_my_forgotten_problems_number_theory
• various_1999_some_pauls_favorite_problems
• various_1999_some_pauls_favorite_problems / problem_1_22