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Claim. For every $\varepsilon>0$ and all large $N$, the function $h(N)$ of Problem 160 satisfies

$$h(N)\le N^{1/3+\varepsilon},$$

that is, $h(N)\le N^{1/3+o(1)}$: the first $N$ integers can be colored with that many colors so that every four-term arithmetic progression carries at least three colors. This lowers the upper exponent $\log3/\log22\approx0.355$ that the site's commentary records to $1/3$. The route, as the claim's summary describes it: a product of two digit-based colorings with $N^{o(1)}$ colors each leaves only four-term progressions whose first and fourth terms share one color and whose middle terms share another (the pattern $ABBA$) among those receiving at most two colors, and a third factor with $9p$ colors, built from base-$p$ carries and the norm form of $\mathbf F_{p^3}$ over $\mathbf F_p$ with $p=O(N^{1/3})$, excludes that pattern. The claim is posted by Rio Itabe and credits GPT-5.6; the write-up is the PDF of the tagged release v0.2.1-review-candidate of the repository ritabe-dev/ErdosProblem160-OneThirdUpper, dated 2026-07-14, which the claim describes as a review artifact with reproduction instructions and checksums. The repository carries a Lean development under ErdosProblems/E160 at the pinned commit, which the release notes of the tagged release say proves the bound $h(N)\le N^{1/3+\varepsilon}$ for every real $\varepsilon>0$ through a chain of theorems; the claim's notes on the tab say only that the release carries reproduction instructions and checksums and that the Lean development includes a bridge between its coloring convention and Erdős's partition convention. The claim itself states that the manuscript has not been independently reviewed and makes no claim of priority or novelty. This corpus has not reviewed the write-up and has not built or audited the Lean development.

Submission note. Posted to erdosproblems.com as a proof claim by Rio Itabe (account ritabe) on 15 July 2026, giving "GPT-5.6" as the AI used:

The claimed partial result is

$$h(N)\leq N^{1/3+o(1)},$$

which would improve

the exponent $\frac{\log 3}{\log 22}$ currently recorded on this page. Two $N^{o(1)}$-colour digit filters force every four-term progression receiving at most two colours from their product to have the proper $ABBA$ pattern. A $9p$-colour carry-and-norm factor over $\mathbf F_{p^3}$, with $p=O(N^{1/3})$, excludes that remaining pattern. Notes: Tagged review release with reproduction instructions and checksums: https://github.com/ritabe-dev/ErdosProblem160-OneThirdUpper/releases/tag/v0.2.1-review-candidate The Lean development includes an exact bridge between the maintained colouring convention and Erdős's original partition convention. The manuscript has not been independently reviewed. No claim of priority or novelty is made.

Covers. An upper bound only: $h(N)\le N^{1/3+o(1)}$. The claim gives no lower bound and does not determine the order of $h(N)$, which is what the problem asks for; the best lower bound recorded on the problem page is $h(N)\gg\exp(c(\log N)^{1/6-o(1)})$. A later claim, [[problems/additive_combinatorics/E0160/claims/2026_07_22_shi_dong|Shi and Dong's bound $N^{1/4+o(1)}$]], asserts a smaller exponent by a different construction.

Depends on. Nothing in this wiki: the construction is self-contained as the summary describes it.

Standing. Claimed. The site's label is OPEN and its page, last edited 2 December 2025, does not mention the claim; the curator records no acceptance. The one comment on the claim thread, by the submitter of the later claim (2026-07-24), says that the one-third argument predates their own and that their preprint will cite it. The Lean development is third-party Lean this corpus has not built, so it gives no formalized evidence; the formal-conjectures file for the problem is a statement, not a proof of this bound. There is no journal publication or arXiv version.