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Statement

Notation (pp. 13--14). A subset of an Abelian group is sum-free when no $a,b,c$ in it, not necessarily distinct, satisfy $a+b=c$. For a set $B$, $s(B)$ is the largest size of a sum-free subset of $B$; for a sequence $A$, $s(A)$ is the largest length of a subsequence whose set of values is sum-free (see Proposition 1.2 for the definition in full).

Theorem 1.3 (p. 14, quoted). "For any finite Abelian group $G$, every set $B$ of non-zero elements of $G$ satisfies $s(B)>\frac27|B|$. The constant $\frac27$ is best possible. Similarly, every sequence $A$ of non-zero elements of $G$ satisfies $s(A)>\frac27|A|$, and the constant $\frac27$ is optimal."

"Best possible" is shown by a family, not by a single group (p. 21): for $G=\mathbb Z_7^s$ and $B=G\setminus\{0\}$, $|B|=7^s-1$ and $s(B)=2\cdot7^{s-1}$, so $s(B)/|B|=2\cdot7^{s-1}/(7^s-1)$ exceeds $\frac27$ and tends to it as $s\to\infty$. No constant larger than $\frac27$ holds in every finite Abelian group, while each fixed group may admit a larger one.

The paper introduces the theorem as settling, for finite Abelian groups, the problem of Babai and Sós of estimating the largest sum-free subset of $n$ elements of a general group (p. 14), and calls it the paper's main result. The abstract (p. 13) states the set case.

Related statements in Section 4. For particular groups the constant improves (pp. 23--24): $s(B)\ge\frac{k+1}{3k+1}|B|$ for every sequence $B$ of nonzero elements of $\mathbb Z_p$ with $p=3k+2$ prime, $s(B)\ge\frac13|B|$ for $\mathbb Z_p$ with $p\equiv1\pmod3$ prime, $s(B)\ge\frac{2^{s-1}}{2^s-1}|B|$ for $\mathbb Z_2^s$, each stated best possible, $s(B)\ge\frac13|B|$ in $\mathbb Z_n$ when $n$ has no prime divisor congruent to $2$ modulo $3$, and Proposition 4.3 (p. 23): "For any prime $p\equiv2\pmod3$ and any $s\ge1$, every sequence $B$ of non-zero elements of the cyclic group $Z_{p^s}$ satisfies $s(B)>\frac13|B|$", whose proof the paper omits (p. 24). Proposition 4.2 (p. 22), obtained by applying Theorem 1.3 (and Proposition 4.1) repeatedly, states that any set of $n$ nonzero elements of a finite Abelian group, and any set of $n$ nonzero reals, can be partitioned into $O(\log n)$ sum-free subsets. Item 5 (p. 24) extends the lower bound $\frac27$, with the same optimal constant, to weakly sum-free sets, which exclude $a_1+a_2=a_3$ only for distinct $a_1,a_2,a_3$.

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, as described on the source card: the statement on p. 14, the proof in Section 3, pp. 18--21, the Section 4 statements on pp. 22--24.

Read depth. Claims checked: the statement, the optimality example and the Section 4 statements listed above were read clause by clause on the page images. The proof of the lower bound (pp. 18--20) was read for structure; the table of Lemma 3.1, whose proof the paper omits, was not recomputed. Nothing here is independently reviewed.

Proof pointer

Section 3, pp. 18--21. The lower bound is proved for sequences, which gives sets. In $\mathbb Z_n$ take the sum-free sets $I_1=\{x:\frac13n<x\le\frac23n\}$ and $I_2=\{x:\frac16n<x\le\frac13n\text{ or }\frac23n<x\le\frac56n\}$. Lemma 3.1 (pp. 18--19) tabulates $|dZ_n\cap I_j|/|dZ_n|$ for the subgroups $dZ_n$, by $n/d$ modulo $6$, and gives the weighted inequality (the paper's (3.1))

$$\frac47\,\frac{|dZ_n\cap I_1|}{|dZ_n|}+\frac37\,\frac{|dZ_n\cap I_2|}{|dZ_n|}\ge\frac27 .$$

Embed $G$ in $\mathbb Z_n^s$, map the terms $b_i$ to $\mathbb Z_n$ by a uniformly random homomorphism $x\mapsto\sum_jx_jb_{ij}$, whose image for each $b_i$ is uniform on a subgroup $d_iZ_n$ with $d_i<n$, and compare the expected numbers $M_1,M_2$ of terms landing in $I_1,I_2$. The zero homomorphism lands no term in either set, so some homomorphism lands more than the average, and $s(B)>M_j$ for $j=1,2$; then (3.1) gives $s(B)>\frac47M_1+\frac37M_2\ge\frac27|B|$ (p. 20). Optimality comes from Theorem 3.2 (p. 21), due to Rhemtulla and Street.

Dependencies

• Theorem 3.2 (p. 21), cited from A. H. Rhemtulla and A. P. Street, Maximum sum-free sets in elementary Abelian $p$-groups, Canad. Math. Bull. 14 (1971), 73--80: for a prime $p=3k+1$ and $G=\mathbb Z_p^s$, the largest sum-free subset of $G$ has $kp^{s-1}$ elements. With $p=7$ it gives the optimality example.
• Lemma 3.1 (pp. 18--19), stated with its proof omitted as an easy case analysis.

Bears on

• Problem 792: the problem concerns sets of integers and the theorem does not bound its $f(n)$. The problem page uses the theorem to test a remark of Erdős's 1965 paper that his bound $n/3$ holds in any finite Abelian group: by the optimality example no constant above $\frac27$, and so not $\frac13$, holds in every finite Abelian group.