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Statement

Notation as in Proposition 1.2: for a sequence $B$, $s(B)$ is the largest length of a subsequence whose set of values has no $a,b,c$, not necessarily distinct, with $a+b=c$.

Proposition 4.1 (p. 21, quoted). "For any sequence $B$ of non-zero reals, $s(B)\ge\frac13|B|$." The paper attributes the statement for sets $B$ to Erdős, in his 1965 paper (P. Erdős, Extremal problems in number theory, Proc. Sympos. Pure Math. VIII, Amer. Math. Soc. (1965), 181--189; Theorem 2 there), and says its proof is similar to that of Proposition 1.2.

Proposition 4.1' (pp. 21--22, quoted from p. 21). "For any sequence $B$ of non-zero reals, $s(B)>\frac13|B|$."

For a set of $n$ nonzero reals this gives a sum-free subset of at least $(n+1)/3$ elements, since the size is an integer.

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: Section 4, item 1, Propositions 4.1 and 4.1' on p. 21, the proof of Proposition 4.1' on pp. 21--22.

Read depth. Claims checked: both statements and the attribution were read clause by clause on the page images. The proof was read and its steps followed, with the observation under Proof pointer; nothing here is independently reviewed.

Proof pointer

Pp. 21--22. Given nonzero reals $b_1,\dots,b_n$, the paper finds integers $c_1,\dots,c_n$ with the same sign pattern on every signed sum: for each $\epsilon\in\{\pm1,0\}^n$, $\sum_i\epsilon_ic_i$ has the sign of $\sum_i\epsilon_ib_i$. The $3^n$ sign conditions, each an equation or a non-strict inequality with a rational bound, form a linear program with rational coefficients that $(b_1,\dots,b_n)$ satisfies, so it has a rational solution, and clearing denominators gives the $c_i$. The paper concludes that no $c_i$ is zero and $s(B)=s(C)$, and Proposition 1.2 gives $s(C)>\frac13n$.

An observation made here on the printed argument: sign conditions with coefficients in $\{\pm1,0\}$ preserve every relation $b_i+b_j=b_k$ with $i\ne j$, but not a relation $2b_i=b_k$, which the paper's sum-free condition also forbids. For $B=(1,2)$ the integer sequence $C=(1,3)$ meets every printed sign condition, yet $s(B)=1$ and $s(C)=2$. What the proof needs is $s(B)\ge s(C)$, and it holds when the linear program also carries the sign conditions with coefficients in $\{0,\pm1,\pm2\}$, which are rational and satisfied by $(b_1,\dots,b_n)$ in the same way; the statement is unaffected.

Dependencies

• Proposition 1.2 (p. 14), applied to the integer sequence $C$.

Bears on

• Problem 792: Erdős posed the question for $n$ real numbers different from $0$; for sets of $n$ nonzero reals this proposition gives a sum-free subset of at least $(n+1)/3$ elements, the bound of Proposition 1.1 in Erdős's real formulation. It gives no upper bound.