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Claim. E. G. Straus, On a problem in combinatorial number theory, J. Math. Sci. 1 (1966), 77--80, cited as [St66] on the problem page: an admissible subset of $\{1,\ldots,N\}$, one in which two sums of distinct elements with different numbers of summands never coincide, has at most $(4/\sqrt3+o(1))\sqrt N$ elements. The paper is not held and has no DOI, and the theorem is stated as two later sources give it. Erdős, Nicolas and Sárközy (Sém. Théor. Nombres Bordeaux (2) 3 (1991), 55--72) state Straus's Theorem 2 as their Lemme 1 (p. 56), $P(\mathcal A,k)\ge k(|\mathcal A|-k)+1$ for the number $P(\mathcal A,k)$ of integers that are sums of exactly $k$ distinct elements of $\mathcal A$, and use Straus's Theorem 4 in the form of their Lemme 2 (p. 57), $F(N)<\frac4{\sqrt3}N^{1/2}+1$ for the largest size $F(N)$ of an admissible subset of $\{1,\ldots,N\}$, proved there from Lemme 1 by following Straus's proof; Deshouillers and Freiman (Astérisque 258 (1999), p. 141) report Straus's bound as $(4/\sqrt3+o(1))\sqrt N$. Every admissible subset of the witness $A=\{1,\ldots,n\}$ obeys this bound, so in the notation of Problem 789

$$h(n)<\frac4{\sqrt3}n^{1/2}+1,$$

a one-line deduction the problem page records. Library result page of the 1991 reproof: Lemme 2.

Covers. The upper bound $h(n)\ll n^{1/2}$. Not covered: the order of growth of $h(n)$, which lies between this bound and Choi's $(n\log n)^{1/3}$, and the sharper constant of Deshouillers and Freiman's Theorem 1, $h(n)\le2\sqrt{n+1/4}-1$ for $n\ge N_0$.

Depends on. Lemme 2 of Erdős, Nicolas and Sárközy, the form of the bound in which this corpus reads it.

Acceptance. Refereed: the paper appeared in the Journal of Mathematical Sciences (Delhi), volume 1 (1966), as its zbMATH record (Zbl 0149.28503) gives it; 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 $h(n)\ll n^{1/2}$ to Straus 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. The theorem is stated from the two later sources; the proof of Lemme 2 has no independent review.