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Claim. The displayed question of Problem 817, whether $g_3(n)\gg3^n$, has a negative answer:

$$\liminf_{n\to\infty}\frac{g_3(n)}{3^n}=0,$$

so no constant $c>0$ and no $n_0$ make $g_3(n)\ge c\,3^n$ for every $n\ge n_0$ (Corollary 1.2). The preprint's Theorem 1.1 is the quantitative form: for every $\varepsilon>0$ there is an integer $d\ge2$ with $g_3(d\ell)\le\varepsilon3^{d\ell}$ for all large $\ell$; the integer $d$ depends on $\varepsilon$, and for a fixed $d$ the construction gives only a constant multiple of $3^{d\ell}$. The argument has two inputs. The first is Korsky's characterization, recorded on the card Corollary 4.2 and on the claim page of Korsky: the subset sums of an $n$-element set 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 whose ternary coefficient sums are injective. The second is a consequence of the construction that disproved Problem 1, which the site credits to GPT-6 Astra, run by Epoch AI, and which the tab summary calls the OpenAI construction: for every $K>0$ there are $n$, $R$, $D$ and integer coefficients $a_0(t),\ldots,a_n(t)$, positive for large $t$ and asymptotic to $Dt^n$, with $KD<R^n$, such that $(x_0,\ldots,x_n)\mapsto\sum a_i(t)x_i$ is injective on the box $\{0,\ldots,Q-1\}^{n+1}$ whenever $Q\le(t-E)R$ for a fixed $E$. With $Q=3^\ell$ the set $\{3^ja_i(t):0\le i\le n,\ 0\le j<\ell\}$ has $(n+1)\ell$ elements and injective ternary sums, by uniqueness of base-three expansions, and its maximum is $(D/(3R^n)+o(1))3^{(n+1)\ell}$, which is below $\varepsilon3^{(n+1)\ell}$ once $K>1/(3\varepsilon)$. The claimant is Simone Costa; the preprint's acknowledgments and the tab entry say that ChatGPT (OpenAI, GPT-5.6 Sol) helped to find and check the base-three application and to revise the manuscript, and that the author checked the statements and references and takes responsibility for the proof. The preprint is on Zenodo (record of 2026-09-05) and on arXiv (v1 of the same day), both linked above. In a comment of 7 September 2026 on the claim's thread, linked above, the claimant announced that the Zenodo record's second version (2026-09-07) adds a self-contained Lean 4 verification that starts from the formal-conjectures statement Erdos817.erdos_817, which formalizes only the displayed question, concludes answer(False), and adapts the parts of the Lean proof for Problem 1 that the construction needs; the record's README is said to carry the provenance.

Submission note. Posted to erdosproblems.com as a proof claim by Simone Costa (account enomis_costa88) on 5 September 2026, giving "ChatGPT (OpenAI, GPT-5.6 Sol)" as the AI used:

I have just posted a preprint giving a negative answer to the question in Problem 817. More precisely, I prove

$$> \liminf_{n\to\infty}\frac{g_3(n)}{3^n}=0. >$$

The argument combines Korsky's characterization in terms of ternary coefficient sums with a consequence of the OpenAI construction for Erdős Problem 1, followed by a base-three expansion. Preprint: https://doi.org/10.5281/zenodo.22313501 ChatGPT (OpenAI, GPT-5.6 Sol) was used in developing and checking the argument and in revising the manuscript; the full AI declaration is included in the paper. I have checked the mathematical statements and references and take responsibility for the proof.

Covers. The displayed question only, in three statements: $g_3(n)\gg3^n$ is false; $\liminf_{n\to\infty}g_3(n)/3^n=0$; and for every $\varepsilon>0$ there is an integer $d\ge2$ with $g_3(d\ell)\le\varepsilon3^{d\ell}$ for all large $\ell$. The claim does not estimate $g_k(n)$ for any $k$ and does not determine the order of $g_3(n)$, whose lower bound in force is $(\sqrt3/(2\sqrt\pi)+o(1))3^n/\sqrt n$ (Korsky, claimed; refereed, $g_3(n)\gg3^n/n$ of Erdős and Sárközy). With the elementary monotonicity $g_3(n+1)\le3g_3(n)$, recorded as a checked observation on the problem page, the claim gives $g_3(n)=o(3^n)$ for all $n$; the quantitative upper bound $g_3(n)\ll3^n/n^{1/3}$ sketched in the thread is recorded on the problem page, not here.

Depends on. The disproof of Problem 1, from whose exposition the preprint's Proposition 2.2, the injective linear form above, is extracted; Korsky's characterization is the paper's own Proposition 4.1, recorded on Korsky's claim page and on the library card.

Standing. Claimed. The site's label was OPEN on 2026-09-18, 2026-10-06 and 2026-10-07, and its commentary does not mention the claim. The tab labeled the claim full on 2026-09-05 and partial from 2026-09-06: a moderator's note inside the claimant's comment of 6 September 2026 calls the full-or-partial question debatable, as the exact wording on the site often is, and records the change to a partial claim, after the claimant had written that they regard the displayed question as the main one. The thread holds four comments: Korsky's thanks of 5 September; a commenter's remark of 6 September that the argument looks correct, later edited to add that it is not a full solution because Erdős asked for an estimate and the Problem 1 construction will not close the gap, which is an informal reading and not a review; the claimant's reply of 6 September with the moderator's note; and the claimant's Lean announcement of 7 September. The proof is followed at the level of its steps on the problem page and is not reviewed in this corpus. No journal publication and no outside review is known. The Lean archive of the second Zenodo version is third-party Lean that this corpus has not built or audited, so it gives no formalized evidence; the formal-conjectures file for the problem is a statement, not a proof.