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Sárközy and Szemerédi prove, as the Satz of their 1965 note, that for distinct positive reals $0<a_1<\cdots<a_n$ and $f(t)$ the number of solutions of $\sum_i\varepsilon_ia_i=t$ with $\varepsilon_i\in\{0,1\}$, every $\varepsilon>0$ has an $n_0(\varepsilon)$ with

$$\max_{t\ge0}f(t)<(1+\varepsilon)\frac8{\sqrt\pi}\cdot\frac{2^n}{n^{3/2}} \qquad(n>n_0(\varepsilon)).$$

This removes the factor $(\log n)^{3/2}$ from the Erdős--Moser bound and is sharp in order, since $a_i=i$ gives $\max_tf(t)>c_32^n/n^{3/2}$. For the first question of Problem 362: an $N$-element set $A\subseteq\mathbb N$ is a set of distinct positive reals, so the number of $S\subseteq A$ with $\sum S=t$ is below $(1+\varepsilon)(8/\sqrt\pi)2^N/N^{3/2}$ once $N>n_0(\varepsilon)$, and the trivial bound $2^N$ covers the finitely many smaller $N$, so the count is $\ll2^N/N^{3/2}$ with an absolute constant (the problem page's authored line). The exact maximum over $N$ distinct positive reals, the middle coefficient of $(1+q)(1+q^2)\cdots(1+q^N)$, attained by $\{1,\ldots,N\}$, and the maximum over all sets of $N$ distinct reals, attained by $\{-\lfloor(N-1)/2\rfloor,\ldots,\lfloor N/2\rfloor\}$, are Stanley's Corollaries 5.1 and 5.3 of 1980, recorded on the problem page as the extremal sets rather than as a settling claim.

The paper's library card is Sárközi and Szemerédi 1965, with a result page for the Satz. The statement is checked against the paper; the indirect proof, through a Sperner-type lemma, was followed for structure only, and nothing here is independently reviewed.

Covers. The first question of Problem 362: for every $N$-element $A\subseteq\mathbb N$ and every $t$, the number of subsets of $A$ with sum $t$ is $\ll2^N/N^{3/2}$. It does not address the second question, the count with the subset size $l$ also fixed, which is Halász's page.

Formalization. The statement file of the formal-conjectures project (FormalConjectures/ErdosProblems/362.lean) states the first question as its declaration erdos_362, marks it solved and points, through its formal_proof attribute, at a Lean 4 file in Boris Alexeev's lean-proofs repository, linked above at the commit the attribute pins. That file declares itself a formalization of a solution to Problem 362, names András Sárközy, Endre Szemerédi and Gábor Halász as its informal authors and Codex and GPT-5.6 Sol as its formal authors, and cites this paper for the first estimate. Its theorem erdos_362 states both estimates of the problem as a conjunction, with absolute constants over all nonempty finite $A\subseteq\mathbb N$; the first conjunct, the bound $C\,2^N/N^{3/2}$ on the number of subsets of $A$ with a given sum, is this claim, and the second is the claim of Halász's page. The file contains no sorry, no axiom command and no native_decide at the pinned commit. This project has not built the file or audited its statement against the problem, so it supplies no formalized evidence; the site's label PROVED (LEAN) refers to this development.

Acceptance. The paper is refereed: A. Sárközy and E. Szemerédi, Über ein Problem von Erdös und Moser, Acta Arith. 11, no. 2 (1965), 205--208, received 26 November 1964; the paper prints the first author as Sárközy, while the site's key and Erdős's 1973 survey write Sárközi. Erdős's surveys of 1973 and 1980 report the theorem as the proof of his conjecture with Moser. The site's curator, Thomas F. Bloom, marks Problem 362 proved and credits this paper with the affirmative answer to the first question. The page is dated by the first day of the publication year, since the paper's first posting carries no finer date.

Depends on. Sárközi and Szemerédi 1965, the Satz.