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Statement

For a set $\mathcal A$ of positive integers, write $s^\wedge\mathcal A$ for the set of all sums of $s$ distinct elements of $\mathcal A$; $\mathcal A$ is admissible when $s^\wedge\mathcal A\cap t^\wedge\mathcal A=\emptyset$ whenever $s\ne t$ (printed p. 141). The paper says the notion "has been introduced by P. Erdős in 1962 (cf. [2]) and called admissibility by E.G. Straus in 1966 (cf. [5])".

Theorem 1 (printed p. 142). "There exists an integer $N_0$, effectively computable, such that for any integer $N\ge N_0$ and any admissible subset $\mathcal A\subset[1,N]$ we have

$$\operatorname{Card}\mathcal A\le2\sqrt{N+1/4}-1.$$

"

The introduction (p. 141) records: Erdős's conjecture that the largest size of an admissible subset of $[1,N]$ is attained by a block of consecutive integers ending at $N$; Straus's computation that $\{N-k+1,N-k+2,\ldots,N\}$ is admissible if and only if $k\le2\sqrt{N+1/4}-1$; Straus's inequality $|\mathcal A|\le(4/\sqrt3+o(1))\sqrt N$; the slight reduction of the constant by Erdős, Nicolas and Sárközy (Théorème 1); and the authors' own $|\mathcal A|\le(2+o(1))\sqrt N$ from part 1 (Israel J. Math. 92 (1995), 33--43), filed as deshouillers_1995_additive_problem_erdos_straus; its Theorem 1, $\operatorname{card}\mathcal A\le2N^{1/2}+CN^{5/12}$, is on printed p. 34 (PDF p. 2), read there clause by clause on the page image and paged on theorem_1. Theorem 1 therefore gives, for $N\ge N_0$, the exact value $\max|\mathcal A|=\lfloor2\sqrt{N+1/4}-1\rfloor$, attained by Straus's block: the block gives the lower bound and Theorem 1 the matching upper bound. This one-line combination is made here; the paper states the theorem and the block computation separately.

Theorem 2 (p. 142, quoted from part 1). "Let $\mathcal A$ be an admissible set included in $[1,N]$, such that $\operatorname{Card}\mathcal A>1.96\sqrt N$. If $N$ is large enough, there exist $\mathcal C\subset\mathcal A$ and an integer $q$ having the following properties : (i) $\operatorname{Card}\mathcal C\le10^5N^{5/12}$, (ii) for some $t$ the set $t^\wedge\mathcal C$ contains at least $3N^{5/6}$ terms in an arithmetic progression modulo $q$, (iii) $\mathcal A\setminus\mathcal C$ is included in an arithmetic progression modulo $q$ containing at most $N^{7/12}$ terms."

Remark (p. 142). The authors state, without giving details, that their method also describes the admissible subsets of $[1,N]$ of largest size: for $N=n^2$ or $N=n^2+n$ with $n$ large enough, the Erdős–Straus block is the unique admissible subset of $[1,N]$ of largest size.

Source. J.-M. Deshouillers and G. A. Freiman, On an additive problem of Erdős and Straus, 2, in Structure theory of set addition, Astérisque 258, Soc. Math. France (1999), 141–148 (the Numdam record, gives MR 1701192 and Zbl 0979.11005; the article's own DOI is 10.24033/ast.442, per its Crossref record read); the copy read is the Numdam file, 9 pages, printed p. $n$ on PDF p. $n-139$. The introduction and Theorems 1 and 2 with the remark on printed pp. 141–142 (PDF pp. 2–3), read on the page images.

Read depth. Claims checked: the definition, the historical account, Theorem 1, Theorem 2 and the remark were read clause by clause on the page images. The proof (Sections 1–3, pp. 142–147) was read for its structure only; $N_0$ is not made explicit in the paper.

Proof pointer

Section 1 (pp. 142–143) proves Proposition 1, a local lemma: for integers $r,s,t,a,q$ with $t\ge2s-q$, $s\ge4r+3+q$ and $0\le a<q$, if $\mathcal D$ is a set of $t$ integers congruent to $a$ modulo $q$ spanning $(t-1+r)q$, then among any $2r+1$ consecutive integers congruent to $sa$ modulo $q$ in the range of $s^\wedge\mathcal D$, at least $r+1$ lie in $s^\wedge\mathcal D$. Section 2 (pp. 144–145) proves Theorem 3, the structure of an admissible $\mathcal A\subset[1,N]$ with $|\mathcal A|=2N^{1/2}+O(N^{5/12})$: the modulus $q$ of Theorem 2 is $O(N^{5/12})$, and, for $\mathcal A=\{a_1<\cdots<a_{|\mathcal A|}\}$, some $u\in[N^{11/24},2N^{11/24}]$ has $a_{|\mathcal A|-u}-a_{u+1}=q(2N^{1/2}+O(N^{11/24}))$, a span estimate for the middle elements. Section 3 (pp. 146–147) takes $\mathcal A$ of maximal cardinality and applies Proposition 1 with $s=\sigma$ and $s=\sigma+q$, where $\sigma=[(t-q)/2]$, to the middle part $\mathcal D=\mathcal A\cap[a_{u+1},a_{|\mathcal A|-u}]$ of $t=|\mathcal A|-2u$ elements, $u$ as in Theorem 3, and derives Theorem 1 in the form $(|\mathcal A|+1)^2\le4N+1$. Not reconstructed here.

Dependencies

Theorem 2 of the authors' first paper (Israel J. Math. 92 (1995), 33--43, DOI 10.1007/BF02762069), quoted as Theorem 2 here; the paper is filed as deshouillers_1995_additive_problem_erdos_straus, and its Theorem 2 is on printed p. 34 (PDF p. 2), read there clause by clause on the page image on 2026-09-22 and paged on theorem_2; the quotation above matches it apart from "modulo $q$" for the original's "with difference $d$". Straus's block computation (J. Math. Sci. 1 (1966), 77–80, not held), quoted on p. 141.

Bears on

• Problem 874: the status-defining source. The problem's $k(N)$ is the largest admissible subset of $\{1,\ldots,N\}$; Theorem 1 with Straus's block computation gives $k(N)=\lfloor2\sqrt{N+1/4}-1\rfloor$ for $N\ge N_0$, hence $k(N)=2N^{1/2}+O(1)$ and $k(N)\sim2N^{1/2}$, the affirmative answer to the site's question; the uniqueness remark is the site's "in some cases the largest such $A$ has the form $(N-k,N]\cap\mathbb N$".
• Problem 875: for an infinite admissible $A=\{a_1<a_2<\cdots\}$, Theorem 1 applied to $A\cap[1,x]$ gives $A(x)\le2\sqrt{x+1/4}-1$ for $x\ge N_0$, so $a_n\ge n(n+2)/4$ for large $n$, and a gap bound $a_{n+1}-a_n\le n^c$ for all large $n$ forces $c\ge1$; both are deductions made here from the theorem.
• Problem 789: the paper's introduction attests Straus's $(4/\sqrt3+o(1))\sqrt N$ bound and the naming of admissibility; the problem's $h(n)$ is bounded by the largest admissible subset of $\{1,\ldots,n\}$.