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Claim. R. Shi and Y. Dong, An improved upper bound for colorings without symmetrically colored $k$-term arithmetic progressions, arXiv:2607.20752 (v1 22 July 2026, v2 28 July 2026, 7 pages), state the following. Call a nontrivial $k$-term progression $a,a+d,\ldots,a+(k-1)d$ symmetrically colored by $c$, for even $k\ge4$, when $c(a+(i-1)d)=c(a+(k-i)d)$ for every $i\le k/2$. For every even $k\ge4$ and every prime $p>k$ there is a coloring of $\mathbb Z/p^{k^2/4}\mathbb Z$ with $O_k(p)$ colors and no symmetrically colored $k$-term progression, hence a coloring of $\{1,\ldots,N\}$ with $O_k(N^{4/k^2})$ colors; for $k=4$ this lowers the exponent of Deng, Tidor and Zhao's $O(N^{\log_{22}3})$-coloring to $1/4$. The construction combines a carry-controlling coloring of the base-$p$ digits with a layered norm map from the extension $\mathbf F_{p^3}$ of $\mathbf F_p$ and a linear term. Taking the product with Behrend-style colorings, the paper claims

$$h(N)\le N^{1/4+o(1)}$$

for the function of Problem 160. The abstract also claims $\rho_4(\alpha)=O_\varepsilon(\alpha^{5-\varepsilon})$ for every $\varepsilon>0$, toward a question of Ruzsa, and that the $k$-progression result disproves a conjectured lower bound of Gowers for all even $k\ge6$; neither bears on Problem 160. The site's proof claim, posted by the account ruizshi on 2026-07-24 and credited to GPT 5.6 Sol, says that the $N^{o(1)}$ question stays open and names the preprint; its one comment, of the same day, outlines the construction. This account follows the arXiv abstract and the claim thread.

Submission note. Posted to erdosproblems.com as a proof claim by Ruizhe Shi, Yiqi Dong (account ruizshi) on 24 July 2026, giving "GPT 5.6 Sol" as the AI used:

We are not able to prove the $N^{o(1)}$ result. But this new result (https://arxiv.org/pdf/2607.20752) can improve the exponent in the upper bound from $\log_{22} 3$ to $1/4$. The approach is to give a new construction for the coloring problem introduced in https://arxiv.org/pdf/2307.06914. And the basic idea is to use the standard $\mathbb F_{p^3}/\mathbb F_p$ norm plus a linear term to color $\mathbb Z/p^4 \mathbb Z$. Then we can take a product of such a coloring with the Behrend-style coloring as previous comment suggested.

Covers. An upper bound only: $h(N)\le N^{1/4+o(1)}$, the smallest exponent claimed for the problem. It gives no lower bound and does not determine the order of $h(N)$; whether $h(N)\le N^{o(1)}$ remains open, as the claim itself says. The earlier claim of [[problems/additive_combinatorics/E0160/claims/2026_07_14_itabe|Itabe's bound $N^{1/3+o(1)}$]] is weaker, and the preprint's authors say on its thread that a future version will cite it.

Depends on. Nothing in this wiki: the construction is self-contained as the abstract describes it, and the Behrend-style product step is the known reduction from symmetric colorings.

Standing. Claimed. The preprint is unrefereed, with no journal record found; the site's label is OPEN and its page, last edited 2 December 2025, does not mention the claim, so the curator records no acceptance. No review of the argument is known here.