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Claim. There is a constant $C$ such that every admissible $A\subseteq\{1,\ldots,N\}$ (a set whose sums of $r$ distinct elements, for distinct $r$, never coincide) has at most $2N^{1/2}+CN^{5/12}$ elements. Straus's block $\{N-k+1,\ldots,N\}$ is admissible for $k=\lfloor2\sqrt N-1\rfloor$, so

$$k(N)=2N^{1/2}+O(N^{5/12}), \qquad\text{hence}\qquad k(N)\sim2N^{1/2},$$

the affirmative answer to the displayed question of Problem 874, with the constant $2$ best possible. The theorem is Theorem 1 of J.-M. Deshouillers and G. A. Freiman, On an additive problem of Erdős and Straus, 1, Israel J. Math. 92 (1995), no. 1--3, 33--43, doi:10.1007/BF02762069, cited as [DeFr95] on the problem page and recorded with its result page on the library card; the abstract states it as "the cardinality of such an admissible subset $\mathcal A$ is at most $(2+o(1))\sqrt N$. As shown by Straus, the constant 2 cannot be improved upon." The proof (Section 6, pp. 41--42) takes $C=10^6$ for large $N$ and deduces the bound from the paper's Theorem 2, the structure theorem for admissible sets with more than $1.96\sqrt N$ elements (result page): two sums of distinct elements with different numbers of summands are forced to coincide once the set is too large. It improves Erdős's $O(N^{5/6})$ of 1962, Straus's $(4/\sqrt3+o(1))\sqrt N$ and the $(143/27)^{1/2}$ of Erdős, Nicolas and Sárközy, none of which answers the asymptotic question. The exact value $k(N)=\lfloor2\sqrt{N+1/4}-1\rfloor$ for all large $N$ is the same authors' 1999 result, recorded on its own claim page, which the site credits. As the library card records, Theorem 1 and the abstract are checked clause by clause and the proof of Theorem 1 from Theorem 2 is read in full; the proof of Theorem 2 is read for structure only, and nothing is independently reviewed.

Depends on. Nothing in this wiki. The lower bound that the asymptotic also needs is Straus's block computation (1966; not held), reported in the paper itself (p. 34) and by Erdős, Nicolas and Sárközy (1991), and recorded on the problem page.

Acceptance. Refereed: Israel Journal of Mathematics 92 (1995), no. 1--3, 33--43, doi:10.1007/BF02762069 (Crossref record read, issue dated February 1995; received March 11, 1993, revised March 22, 1994); the page is named by the issue's month, filled to its first day. Not reviewed: the site's commentary credits the affirmative answer to the authors' 1999 paper, [DeFr99], and does not cite this one, so no curator credit attaches to it; its result is what the 1999 paper's introduction and the site's "proved" label build on.