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Claim. In the notation of Problem 817, with $T_m$ the $m$th central trinomial coefficient, for every $n\ge1$

$$g_3(n)\ \ge\ \frac{T_n-1}{2}+\sum_{j=0}^{n-1}T_j, \qquad\text{hence}\qquad g_3(n)\ \ge\ \Bigl(\frac{\sqrt3}{2\sqrt\pi}+o(1)\Bigr)\frac{3^n}{\sqrt n}$$

(Theorem 1.1, p. 2); for every fixed $k\ge4$, $g_k(n)\gg_k((k-1)/(k-2))^n\,n^{-\log_2((k-1)/(k-2))}$, so $\liminf g_k(n)^{1/n}\ge(k-1)/(k-2)$ (Theorem 1.2, p. 2); and for every $k\ge3$ and prime $p\ge3$, $g_k(n)<2p^{\rho_{p,k}(n)-1}$ with $\rho_{p,k}(n)=\max(q-1,\lceil2n/q\rceil)$ and $q=\min\{p,k\}-1$, so $\limsup g_k(n)^{1/n}\le\min_pp^{2/(\min\{p,k\}-1)}$ (Theorem 1.3, p. 3). S. Korsky, Arithmetic progression-free subset-sum sets, arXiv:2606.24139v1 (23 June 2026), cited as [Ko26] on the problem page. Library home korsky_2026_arithmetic_progression_free_subset_sum_sets; result pages Theorem 1.1, Theorem 1.2, Theorem 1.3 and Corollary 4.2. The $k=3$ bound rests on Proposition 4.1 and Corollary 4.2 (p. 6): the subset sums of $A$ avoid non-trivial three-term progressions exactly when the $3^n$ sums $\sum\varepsilon_aa$ with $\varepsilon_a\in\{0,1,2\}$ are distinct, so $g_3(n)$ is the least possible maximum of $n$ positive integers with injective ternary coefficient sums, and the exact bandwidth of the ternary grid gives the bound. Remark 4.6 (p. 8) tabulates $g_3(n)=1,3,8,22$ for $n\le4$ against the bound's values $1,3,8,21$ and notes $g_3(n)\le3^{n-1}$, leaving a factor of order $\sqrt n$ between the bounds. Theorem 1.2 proceeds by chain expansion and averaging over unused generators, Theorem 1.3 by a carry-free base-$p$ digit construction, and Corollary 1.4 (p. 3) places the logarithms of the lower and upper exponential rates between $(1+o(1))/k$ and $(2+o(1))\log k/k$. The author announced the preprint in the site's discussion thread on 23 June 2026, noting that a second-moment argument already gives $\Omega(3^n/\sqrt n)$ with a worse constant.

Covers. The lower bound $g_3(n)\ge(\sqrt3/(2\sqrt\pi)+o(1))3^n/\sqrt n$, which sharpens the refereed $g_3(n)\gg3^n/n$ of Erdős and Sárközy, and the bounds $(k-1)/(k-2)\le\liminf g_k(n)^{1/n}$ and $\limsup g_k(n)^{1/n}\le\min_pp^{2/(\min\{p,k\}-1)}$ for fixed $k\ge4$. Not covered: the displayed question $g_3(n)\gg3^n$, which the pending claim of Costa answers in the negative, and the order of $g_k(n)$ for any $k$, which stays open; the paper itself says that removing the factor $\sqrt n$ would settle the principal question of Erdős and Sárközy.

Depends on. No page of this wiki; the ternary characterization is the paper's own Proposition 4.1, and the bounds of Erdős and Sárközy are not used in the proofs.

Standing. Claimed. The paper is a preprint with no journal record (Crossref, 2026-09-18). The site's label was OPEN on 2026-09-18 and on 2026-10-07 and its commentary does not mention the paper; the curator's thread reply of 23 June 2026 discusses which question Erdős meant and is not an acceptance. The preprint is cited by the claim of Costa, which takes Corollary 4.2 as one of its two inputs, and by the thread comment of 9 September 2026 that extends its table of exact values. The statements named above are checked clause by clause and the proofs read for structure only; this is not an independent review.