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Claim. Theorem IV of P. Erdős, Számelméleti megjegyzések, III. Néhány additív számelméleti problémáról (Remarks on number theory III), Mat. Lapok 13 (1962), 28--38, printed p. 34, in the corpus's words: if $1\le a_1<a_2<\cdots$ is a sequence of integers in which two sums of distinct terms with different numbers of summands never coincide (condition (1')), then $A(x)<Cx^{5/6}$ for every $x$, with $C$ an absolute constant. The proof (pp. 35--36), with Rényi's form of the large sieve, concerns the terms up to $x$ only and so applies to a finite admissible subset of $\{1,\ldots,n\}$; with that witness, in the notation of Problem 789,

$$h(n)<Cn^{5/6},$$

which is how Erdős's 1965 survey reads it ("It is known that $h(n)<c_8n^{5/6}$ [5]") and how the site states it. The paper's eleven pages also construct an infinite admissible sequence of unspecified polynomial growth, but contain no statement of the form $h(n)\gg(n\log n)^{1/3}$, a lower bound the site's commentary attributes to this paper together with Choi's of 1974; that bound is Choi's alone. Cited as [Er62c] on the problem page. Library home erdos_1962_szamelmeleti_megjegyzesek; result page Theorem IV.

Covers. The upper bound $h(n)<Cn^{5/6}$ only, superseded by Straus's $n^{1/2}$. Not covered: any lower bound for $h(n)$, and the order of growth of $h(n)$.

Depends on. No page of this wiki; the proof rests on the published large sieve inequality.

Acceptance. Refereed: the paper appeared in Matematikai Lapok, volume 13 (1962), with Russian and English summaries (zbMATH record Zbl 0123.25503; no DOI); the volume carries no publication day, so this page is named by the first day of its year. The site's curator, Thomas F. Bloom, credits $h(n)\ll n^{5/6}$ to this paper in the problem page's commentary (label OPEN); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. Theorem IV is stated from printed p. 34 and the English summary; its proof is not reviewed in this corpus.