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Statement

Notation (p. 14). For a subset $B$ of an Abelian group, $s(B)$ is the largest size of a sum-free subset of $B$, sum-free meaning that no $a,b,c$ of the set, not necessarily distinct, satisfy $a+b=c$ (p. 13). For a sequence $A=(a_1,\dots,a_n)$ of elements of an Abelian group, not necessarily distinct, $s(A)$ is the largest $k$ for which there are indices $1\le i_1<\dots<i_k\le n$ such that the set $\{a_{i_1},\dots,a_{i_k}\}$ is sum-free. Repeated terms are counted with multiplicity in $k$, while the sum-free condition is tested on the set of their values.

Proposition 1.2 (p. 14, quoted). "For any sequence $B$ of non-zero integers, $s(B)>\frac13|B|$."

For a sequence of distinct terms this is Proposition 1.1; the paper calls Proposition 1.2 "clearly stronger" (p. 14).

The upper side for sequences (p. 14, proved in Section 2, pp. 16--18). The paper constructs sequences $B$ with $s(B)<\frac{11}{28}|B|$: Corollary 2.4 (p. 18) is the sequence $S=(ijl:1\le i\le2,\ 1\le j\le5,\ 1\le l\le14)$ of $140$ products, with $s(S)\le55$, so $s(S)/|S|\le\frac{11}{28}$. Corollary 2.3 (p. 17) states that for every sequence $A$ of nonzero integers there is a sequence $B$ with

$$\frac{s(B)}{|B|}\le\frac{s(A)}{|A|}-\frac{1}{(|A|-s(A)+1)!\,e\,|A|},$$

so the infimum of $s(B)/|B|$, as $B$ ranges over all sequences of integers, is not attained (p. 14). Remark 2.5 (p. 18) notes that Lemma 2.2 yields sequences with smaller ratios still and omits their computation, since the method does not seem to close the gap between $\frac13$ and $\frac{11}{28}$; on p. 26 the paper names the best-possible constants in Propositions 1.1 and 1.2 as an open question.

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 notation and the proposition on p. 14, the proof on p. 15, Corollaries 2.3 and 2.4 on pp. 17--18.

Read depth. Claims checked: the notation, the proposition, Corollaries 2.3 and 2.4 and the statements of p. 14 were read clause by clause on the page images. The proof on p. 15 was read and its steps followed; the proofs of Lemma 2.2 and Corollaries 2.3 and 2.4 were read for structure only. Nothing here is independently reviewed.

Proof pointer

P. 15. For the terms $b_1,\dots,b_n$ of $B$, take a prime $p=3k+2$ with $p>2\max_i|b_i|$, so that no $b_i$ vanishes modulo $p$, and the interval $C=\{k+1,\dots,2k+1\}$, a sum-free subset of $\mathbb Z_p$ with $|C|/(p-1)=(k+1)/(3k+1)>\frac13$. For $x$ uniform in $\{1,\dots,p-1\}$, each $xb_i\bmod p$ is uniform on the nonzero residues, so the expected number of $i$ with $xb_i\bmod p\in C$ exceeds $n/3$. Fix an $x$ doing at least as well as the expectation: more than $n/3$ terms land in $C$, and they form a sum-free subsequence, since $a+b=c$ among them would give $xa+xb\equiv xc\pmod p$ inside $C$.

Dependencies

None stated. The construction side uses Schur's theorem (Theorem 2.1, p. 16, with $f(2)=5$ and $f(3)=14$) through Lemma 2.2 (pp. 16--17). Proposition 4.1' (pp. 21--22) deduces the same strict bound for sequences of nonzero reals from this proposition.

Bears on

• Problem 792: the problem concerns sets of $n$ integers, where this proposition reduces to Proposition 1.1, the bound $f(n)\ge(n+1)/3$ for sets of nonzero integers. The sequence constructions with ratio below $\frac{11}{28}$ allow repeated terms and give no upper bound for $f(n)$, which is defined over sets.