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Claim. Among any $n$ real numbers different from $0$ there are at least $n/3$ with no relation $a+b=c$ among them, $a=b$ included (Theorem 2 with condition (27), printed p. 186, whose indices satisfy $1\le j_1\le j_2<j_3\le k$). In the notation of Problem 792,

$$f(n)\ge\frac n3,$$

for sets of nonzero integers, the first lower bound for $f(n)$ and the one that, with Eberhard, Green and Manners's upper bound, fixes the main term; the theorem excludes $0$, so for a set of $n$ integers containing $0$ it gives $(n-1)/3$, with the same main term. The proof (pp. 186--187) takes, for each $a_r$, the set of $\alpha\in(0,T)$ with $a_r\alpha\bmod1$ in $(1/3,2/3)$, of measure $T/3$ up to a bounded error, and picks an $\alpha$ lying in at least $n/3$ of these sets; the corresponding $a_r$ satisfy (27) because $(1/3,2/3)$ modulo $1$ contains no sum of two of its points. The paper remarks that the theorem holds in any finite Abelian group; Alon and Kleitman's Theorem 1.3 of 1990 shows the constant $2/7$ is best possible there, so that remark is false as stated, as the problem page records. P. Erdős, Extremal problems in number theory, Proc. Sympos. Pure Math. VIII (Theory of Numbers), Amer. Math. Soc. (1965), 181--189, cited as [Er65] on the problem page. Library home erdos_1965_extremal_problems_number_theory; result page Theorem 2.

Covers. The lower bound $n/3$ for sets of $n$ nonzero integers, $(n-1)/3$ when $0\in A$. Not covered: the second-order term, for which the refereed Bourgain's Proposition 1.3 gives $(n+2)/3$ for sets of $n\ge3$ positive integers and Bedert's preprint (claimed) asserts $c\log\log n$, and the upper bound, which is Eberhard, Green and Manners's $o(n)$.

Depends on. No page of this wiki; the half-page proof is self-contained.

Standing. Claimed. The paper appeared in a Proceedings of Symposia in Pure Mathematics volume, and no evidence that the volume was refereed is on record, so refereed is not listed. The site's curator, Thomas F. Bloom, credits the simple proof of $f(n)\ge n/3$ to Erdős in the problem page's commentary (label OPEN, page last edited 23 January 2026); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. For the site's $f(n)$, $f(n)\ge n/3$ holds for $n\ge2$ by Alon and Kleitman's Proposition 1.1 applied to the nonzero elements, and fails at $n=1$, where $A=\{0\}$ gives $f(1)=0$. The statement and (27) are checked; the proof is followed for its structure only and not checked in this corpus.