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Claim. Theorem 1.3 of Y. Zhi, Branch-Sensitive Density Increments for $k$-Configurations, arXiv:2610.03750v1, p. 3: "Let $c<1/36$. For every sufficiently large finite non-empty set $A\subseteq\mathbb Z$, there is a subset $B\subseteq A$ of size at least $(\log|A|)^{1+c}$ such that $b_1+b_2\notin A$ whenever $b_1,b_2\in B$ are distinct." In the notation of Problem 787,

$$g(n)\ge(\log n)^{1+c}\qquad\text{for every }c<\tfrac1{36}\text{ and all large }n.$$

It is deduced from Theorem 1.1 (p. 3): for a finite abelian group $G$ of odd order, $k\ge2$ and $A\subseteq G$ of density $\alpha>0$, with $L=\log(2/\alpha)$ and $\Lambda=L+\log(2k)$, the proportion of $k$-tuples $(x_1,\dots,x_k)\in G^k$ whose means $(x_i+x_j)/2$, $1\le i\le j\le k$, all lie in $A$ is at least $\exp(-C\Phi)$ with $C$ absolute and $\Phi=k^{36}L^6\Lambda^6+k^{34}L^8\Lambda^5+k^{32}L^{14}\Lambda+k^{30}L^{16}$, where Beker's Theorem 3.1 has $\exp(-O(k^{68}L^{16}))$. The paper reruns Beker's density-increment iteration and keeps apart the outcomes of the graph-counting step (a row-degree anomaly, a non-centered grid anomaly, and a centered grid anomaly treated through the positivity of Gram moments), each charged its own density gain and rank cost. The proof of Theorem 1.3 (pp. 16--17) puts the new bound into Beker's Proposition 4.1, whose threshold $\exp(k^{68+o(1)})$ becomes $\exp(k^{36+o(1)})$, and then follows Sanders's iteration, in which a threshold $\exp(k^{\beta+o(1)})$ gives every $c<1/\beta$. The paper's declaration on the use of artificial intelligence (p. 19) says that ChatGPT (GPT-5.6 Sol) assisted with literature organization, LaTeX drafting and editing, language polishing, and iterative checking of proof details and quantitative bookkeeping, and that the author is responsible for the mathematical content. Cited as [Zh26] on the problem page. The preprint is not held in the library.

Covers. The lower bound $g(n)\ge(\log n)^{1+c}$ for every $c<1/36$, a wider explicit range than Beker's $c<1/68$ and an explicit form of Sanders's bound. Not covered: the order of growth of $g(n)$.

Depends on. Beker's theorem, whose argument the paper refines; Sanders's theorem, whose iteration turns the threshold into the exponent; and Choi's 1971 paper for the reduction from real sets to integer sets, which carries the theorem to the erdosproblems.com formulation; the paper states the problem for sets of integers. The paper cites Beker's preprint without a version. It follows the proof of Beker's Lemma 2.2 up to its final step (see Standing) and the proof of his Proposition 3.2, which its own Proposition 2.7 replaces, and it reuses the cyclic embedding from the proof of his Theorem 1.1 and the frequency propagation from the proof of his Theorem 3.1. It uses his local density increments (Proposition 3.4 and Corollary 3.5), Lemmas A.2, B.7, B.10 and B.11 of his appendices, his Proposition 4.2 and his Lemma 4.3 as stated. It reruns the proof of his Proposition 4.1 with its own Theorem 1.1 in place of his Theorem 3.1 (pp. 16--17). One input is not checked here: Beker's Proposition 4.2 assumes the hypotheses of his Proposition 4.1, among them $|X|\ge\exp(k^{68+o(1)})$, and the paper applies it to sets with $|X|\ge\exp(k^{36+o(1)})$ without comment.

Standing. Claimed. arXiv:2610.03750 has one version, v1, submitted on 25 September 2026 (20 pages), with no journal reference, and a Crossref title query on 2026-10-13 found no record. The erdosproblems.com commentary on the problem (page last edited 23 January 2026, accessed 2026-10-13) does not cite the preprint, and the problem's thread and proof-claim tab, frozen on 6 October 2026, do not mention it. The paper was submitted before Beker's v2 of 2 October 2026, which corrects an error in the proof of Lemma 2.2 of v1 (Lemma 2.3 of v2), as Beker's page records; the paper's "[1, Lemma 2.2]" is that lemma. By the paper's own account, its Proposition 2.2 runs the induction of that proof but keeps the centered output instead of applying the final step, which turns that output into a non-centered grid norm or a row-degree bound; the correction lies in that final step. This is a comparison of the two texts, not a check of either proof. The paper was read in full for its structure; its proofs and the inputs it takes from Beker's preprint are not checked in this corpus.