Eberhard 2014 sets integers no large sum free

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theorem_1_1: The Eberhard–Green–Manners theorem that Erdős's n/3 is asymptotically sharp: sets of n positive integers exist whose every subset of size larger than (1/3 + epsilon) n contains x, y, z with x + y = z, even with x and y distinct.

theorem_6_4: The Eberhard–Green–Manners rough structure theorem for sets of difference doubling below 4: a finite set A of integers with |A - A| at most (4 - epsilon)|A| has density at least 1/2 + c epsilon on some arithmetic progression of length bounded below by a function of epsilon times |A|.

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Eberhard, Sean and Green, Ben and Manners, Freddie, 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 (Crossref record read).

The copy read for this card is arXiv:1301.4579v3 (29 July 2026; 31 pages), a 2026 revision of the 2014 paper whose arXiv comment says it "corrects a very small inaccuracy in Lemma 6.3" (v1 19 January 2013, v2 28 January 2013); its pagination is used here, and the journal text was not compared. Read status: claims checked for the definition of $f(n)$ (sets of $n$ nonzero integers), Theorem 1.1 (p. 2), the stronger distinct-summand form and the subadditivity argument (pp. 1--2), and for Theorem 6.4 with Theorems 6.1, 6.2 and 4.1 and Lemma 6.3 (pp. 9, 21--22), each read clause by clause on the printed pages; the proof of Theorem 1.1 (Sections 3--5 and Appendix A) was not checked, and the proof of Theorem 6.4 (p. 22) was read but not checked step by step. The statements are on theorem_1_1 and theorem_6_4. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1301.4579), every other right reserved.

Theorem 1.1 constructs, for each n, an n-element set of positive integers whose sum-free subsets all have at most n/3 + o(n) elements, so the limit sigma = lim f(n)/n equals 1/3 and Erdos's simple rotation lower bound f(n) >= n/3 is asymptotically optimal; this answers a question of Erdos from 1965 that had only been approached from above (Hilton 7/15, printed "Hinton" in the paper, Klarner 3/7, Alon-Kleitman, Lewko 11/28, Alon's strict improvement). The constructed set is stronger than required: any subset with more than (1/3 + eps)|A| elements has two distinct members whose sum is also a member, which, the paper says, answers a further question asked in Erdos's 1965 paper (p. 2). The proof identifies the local obstructions, coming from Z/QZ (odd elements, residues 2 and 3 mod 5, etc.) and from R (intervals [x,2x)), shows in Section 3 that these are in some sense the only obstructions, and reduces to a local problem, stated roughly as Problem 2.1 (p. 3): a weight function w on Z/QZ x [0,1] such that any open set of w-measure at least 1/3 + eps contains a summing triple, in the stronger form of Proposition 3.1 (p. 5); the set A is then built from w by a random selection (3.3), and the arithmetic regularity and counting lemmata show that it has the property (Theorem 3.3). A separate ingredient of independent interest is a rough structure theorem for sets with small difference set (Theorem 4.1, p. 9; Section 6 derives from it Theorem 6.4, p. 22): a finite set A of integers with |A - A| <= (4-eps)|A| has density at least 1/2 + c*eps on some arithmetic progression P of length >>_eps |A|; the statement leaves c unnamed, and Remark (i) (p. 22) says the argument gives c "something like" 2^-1000. Theorem 1.1 settles the upper-bound side of problem 792, which asks to estimate the largest sum-free subset guaranteed inside any n-element set of integers: the answer is n/3 + o(n).

Source: https://arxiv.org/abs/1301.4579.

Bears on. #792: Theorem 1.1 (p. 2 of arXiv v3) is the site's upper bound $f(n)\le n/3+o(n)$, refereed. The paper says its stronger form with $x\ne y$ "answers a further question asked in [Erd65]"; the problem page reads that question as Erdős's 1965 guess $f(n)=[(n+2)/2]$ for distinct summands, which the stronger form shows false for large $n$. The paper's $f(n)$ is over sets of $n$ nonzero integers, the site's over $A\subset\mathbb Z$; the difference is a shift by one in $n$ when $0\in A$.

Results. Pages and labels are those of arXiv v3 (pp. 1--31).

• Theorem 1.1 (p. 2): for every n some n-element set of positive integers has all its sum-free subsets of size at most n/3 + o(n); equivalently sigma = lim f(n)/n = 1/3.
• Strengthening (p. 2, on the Theorem 1.1 page): the constructed A has every subset of size > (1/3+eps)|A| containing x+y=z with x distinct from y, which the paper says answers a further question asked in Erdos's 1965 paper.
• Theorem 6.4 (p. 22, via Theorem 6.1, p. 21, and Lemma 6.3, p. 22): a finite integer set A with |A-A| <= (4-eps)|A| has density at least 1/2 + c*eps on some arithmetic progression of length >>_eps |A|, with c unnamed. The page also states Theorem 6.1 (A in {1,...,N}, |A-A| <= 4|A| - eps N, density 1/2 + eps/5 on a progression of length >>_eps N) and Theorem 6.2 (the analogue for open subsets of [0,1], density 1/2 + eps/7).
• Theorem 4.1 (p. 9): for every eps > 0 there is delta >>_eps 1 such that every A in {1,...,N} whose set of delta-popular differences has at most 4|A| - eps N elements has |A cap P| >= (1/2 + eps/5)|P| for some arithmetic progression P in {1,...,N} of length >>_eps N; recorded on the Theorem 6.4 page, no page of its own.
• Local problem (Problem 2.1, p. 3, and Proposition 3.1, p. 5): reduces the construction to a weight function w on Z/QZ x [0,1] for which, roughly, any open set of w-measure >= 1/3 + eps contains a summing triple; described on the Theorem 1.1 page, no page of its own.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.