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Claim. Let $K(N)$ be the least $t$ such that every $t$-element subset of $\{1,\ldots,N\}$ has a three-term arithmetic progression among its positive subset sums. Then for $N>N_0$

$$\Bigl[\frac{\log N}{\log3}\Bigr]+2\ \le\ K(N)\ <\ \frac1{\log3}(\log N+\log\log N)+2$$

(Theorem 4, printed p. 251). P. Erdős and A. Sárközy, Arithmetic progressions in subset sums, Discrete Math. 102 (1992), no. 3, 249--264, cited as [ErSa92] on the problem page. Library home erdos_sarkozy_1992_arithmetic_progressions_subset_sums; result page Theorem 4. The paper never writes $g_k(n)$. In the notation of Problem 817, an $n$-element $A\subseteq\{1,\ldots,N\}$ whose subset sums avoid non-trivial three-term progressions gives $K(N)\ge n+1$, so the upper half of the theorem yields $g_3(n)\gg3^n/n$. The interval count in the proof (p. 261), with the factor $2$ that the print omits restored, is that the $3^n$ distinct sums $\sum\varepsilon_aa$ with $\varepsilon_a\in\{0,1,2\}$ lie in $\{0,1,\ldots,2\sum_{a\in A}a\}\subseteq\{0,1,\ldots,2nN\}$; it gives the explicit $g_3(n)\ge(3^n-1)/(2n)$ directly. The print counts the sums in $\{0,1,\ldots,\lvert\mathcal A^*\rvert N\}$, a slip: the corrected count gives the theorem's upper half only with $N$ replaced by $2N$, and the theorem as printed holds for large $N$ by a second-moment count (an observation made here, recorded on the problem page). The result page records these translations, which are not statements of the paper. The site's commentary credits Erdős and Sárközy with $g_3(n)\gg3^n/n^{O(1)}$, leaving the exponent unspecified; the paper's own argument gives exponent $1$. The lower half of the theorem is the set of powers of three, the problem's $g_3(n)\le3^{n-1}$. The paper asks (p. 252) whether $K(N)=\log N/\log3+O(1)$, which is the problem's displayed question $g_3(n)\gg3^n$, and adds that it knows no $N$ with $[\log N/\log3]+2<K(N)$.

Covers. The lower bound $g_3(n)\gg3^n/n$, in the explicit form $g_3(n)\ge(3^n-1)/(2n)$, read through $K(N)$ as the result page records, and the upper bound $g_3(n)\le3^{n-1}$ from the powers of three. Not covered: the estimate of $g_k(n)$ for any $k$, which stays open, and the displayed question $g_3(n)\gg3^n$, which the paper poses and the pending claim of 2026 answers in the negative. The sharper lower bound $(\sqrt3/(2\sqrt\pi)+o(1))3^n/\sqrt n$ is the pending claim of Korsky.

Depends on. No page of this wiki; the proof uses only the uniqueness of ternary expansion and counting.

Acceptance. Refereed: the paper is the publisher's version of record in Discrete Mathematics (Crossref record, as the problem page records; the record gives the issue month, May 1992, and no day, so this page is named by the first day of its year). The site's curator, Thomas F. Bloom, credits the bound to Erdős and Sárközy in the problem page's commentary, with the label OPEN; the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The theorem and the remark that follows it are checked clause by clause and the proof of Theorem 4 followed step by step, with the slip in the count of p. 261 noted above; this is not an independent review.