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Statement

Printed p. 13: "A subset $A$ of an Abelian group is called sum-free if $(A+A)\cap A=\emptyset$, i.e., if there are no (not necessarily distinct) $a,b,c\in A$ such that $a+b=c$." Proposition 1.1 (pp. 13--14): "Any set $B$ of $n$ non-zero integers contains a sum-free subset $A$ of cardinality $|A|>\frac13n$."

Since $|A|$ is an integer, $|A|>n/3$ is $|A|\ge(n+1)/3$, the form the site's Problem 792 quotes. The introduction (p. 13): "We have found a very simple proof of the following statement, which answers this question [Caro's question of a sum-free subset of size $>cn$]. Not surprisingly we learned later that almost the same result, without the strict inequality and with a rather similar proof, had been proved by Erdős more than twenty years ago (see [7])." Proposition 1.2 (p. 14) extends the bound to sequences of nonzero integers, "which is clearly stronger than Proposition 1.1", and Proposition 4.1' (p. 21) to sequences of nonzero reals. On the other side, the construction of pp. 15--16 shows that the constant $\frac13$ cannot be replaced by $\frac{12}{29}$.

Source. N. Alon and D. J. Kleitman, 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; printed pp. 13--14 (PDF pp. 2--3 of the fifteen-page scan, image-only), read on the page images; the site's key [AlKl90] for Problem 792.

Read depth. Claims checked: the definition and Proposition 1.1 were read clause by clause on the page images. The proof (Section 2, p. 15) was read and its steps followed; it is not independently reviewed. Eberhard, Green and Manners (2014, p. 1) credit Alon and Kleitman with pointing out that Erdős's argument can be modified to give $|A|\ge(n+1)/3$, and sketch the modification in Erdős's terms ($A_\theta$ is empty for $\theta\approx0$, so $|A_\theta|>n/3$ for some $\theta$); the paper's own proof is the modular argument under Proof pointer.

Proof pointer

Section 2 of the paper, "simple proofs of Propositions 1.1 and 1.2 which slightly improve Erdős' result" (p. 14). The proof (p. 15) is that of Proposition 1.2, which implies Proposition 1.1. For the elements $b_1,\dots,b_n$ of $B$, take a prime $p=3k+2$ with $p>2\max_i|b_i|$ and the sum-free interval $C=\{k+1,\dots,2k+1\}$ of $\mathbb Z_p$, and choose $x\in\{1,\dots,p-1\}$ uniformly at random. Each $xb_i\bmod p$ lies in $C$ with probability $|C|/(p-1)=(k+1)/(3k+1)>1/3$, so some $x$ puts more than $n/3$ of the $b_i$ in $C$, and those $b_i$ form a sum-free set, since a relation $a+b=c$ among them would give one inside $C$.

Dependencies

• Proposition 1.2 (p. 14), of which this proposition is the case of distinct terms; the paper proves the two together (p. 15).

The argument is a modular counterpart of Erdős's 1965 real-number argument (Theorem 2), which the authors call "a rather similar proof" (p. 13); on p. 21 they call the proof of the real-number statement, their Proposition 4.1, which they say Erdős proved for sets, "similar to that of Proposition 1.2".

Bears on

• Problem 792: the bound $f(n)\ge(n+1)/3$ the site attributes to Alon and Kleitman, for sets of $n$ nonzero integers, the strict-inequality strengthening of Erdős's $f(n)\ge n/3$.