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Claim. There is an effectively computable $N_0$ such that for every $N\ge N_0$ every admissible $A\subseteq\{1,\ldots,N\}$ (a set whose sums of $r$ distinct elements, for distinct $r$, never coincide) has at most $2\sqrt{N+1/4}-1$ elements. Straus's block $\{N-k+1,\ldots,N\}$ is admissible exactly when $k\le2\sqrt{N+1/4}-1$, so for $N\ge N_0$

$$k(N)=\left\lfloor2\sqrt{N+1/4}-1\right\rfloor,$$

hence $k(N)=2N^{1/2}+O(1)$ and $k(N)\sim2N^{1/2}$: the answer to Problem 874 is yes, and the estimate it asks for is exact for all large $N$. The theorem is Theorem 1 of J.-M. Deshouillers and G. A. Freiman, On an additive problem of Erdős and Straus, 2, in: Structure theory of set addition, Astérisque 258, Société mathématique de France (1999), 141--148, cited as [DeFr99] on the problem page and recorded with its result page on the library card. The block computation is Straus's (1966; not held) and is reported in the same paper and by Erdős, Nicolas and Sárközy (1991); combining it with Theorem 1 is the one-line step the problem page records. The paper also remarks that for $N$ of the form $n^2$ or $n^2+n$, $n$ large, the block is the only admissible subset of maximal size.

The proof rests on the structure theorem for admissible sets with more than $1.96\sqrt N$ elements from the authors' first paper (Israel J. Math. 92 (1995), 33--43, its Theorem 2), a local lemma on sums of $s$ distinct elements of a set that nearly fills an arithmetic progression, and a refined structure theorem for admissible sets of size $2N^{1/2}+O(N^{5/12})$. $N_0$ is not made explicit, so the exact formula is proved for large $N$ only; for every $N>1$ it is a conjecture of Erdős, Nicolas and Sárközy, which this result does not settle. The asymptotic $k(N)\sim2N^{1/2}$ itself was first proved by the same authors in 1995, whose Theorem 1, $k(N)\le2N^{1/2}+CN^{5/12}$, has its own claim page; this paper adds the exact value for large $N$. The earlier bounds $\limsup k(N)N^{-1/2}\le4/\sqrt3$ (Straus) and $\le(143/27)^{1/2}$ (Erdős, Nicolas and Sárközy) and Erdős's 1962 bound $k(N)<CN^{5/6}$ are superseded and are not claims about the question as asked.

Depends on. Theorem 2 of Deshouillers and Freiman (1995), the structure theorem the proof quotes; the claim also rests on the block computation stated above.

Acceptance. Refereed: the paper appeared in Astérisque, vol. 258 (1999), pp. 141--148, the Société mathématique de France's Astérisque, whose Crossref record for DOI 10.24033/ast.442 types the article as a journal article in that venue (MR 1701192, Zbl 0979.11005; Numdam and Crossref records read); the volume carries the year only, so this page is named by the first day of 1999. Reviewed: the site's curator, Thomas Bloom, marks the problem proved on erdosproblems.com and credits Deshouillers and Freiman with proving the conjecture for every large $N$; the curator's acceptance is the documented acceptance. Read depth: Theorem 1, Theorem 2 and the uniqueness remark are checked clause by clause; the proof is read for structure only; nothing is independently reviewed, so no further evidence is listed.