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Claim. Przemyslaw Chojecki, Consecutive-Sum Representations: A Random-Block Construction with Deterministic Repair, dated 11 February 2026 and posted to the site's discussion thread of Problem 358 that day. The abstract claims a probabilistic construction of an infinite increasing set $A$ with $f(n)\ge c\log n$ for all sufficiently large $n$, hence $f(n)\to\infty$, which would answer both questions yes. The natural numbers are cut into consecutive blocks of length about $x^{\alpha}$ at $x$; the odd-numbered (red) blocks take each integer independently with probability $1/2$, and the even-numbered (blue) blocks are then filled deterministically to repair the integers left with too few representations. The post says that the write-up was produced with GPT-5.2 Pro, and that Gemini and Grok both judged the argument correct. It follows the strategy of red and blue blocks that Terence Tao proposed on the thread on 7 February 2026.

Submission note. Posted to the site's forum by Przemyslaw Chojecki on 11 February 2026:

After some back and forth with GPT-5.2 Pro I managed to get this write-up for the argument with red/blue coloring (red probabilistic, blue fixing). Also I've run it through Gemini+Grok and both models agree that the argument is correct. This is basically taking Terence Tao strategy described below and doing the bookkeeping.

Rejection. On 12 February 2026 Tao replied in the thread that ChatGPT Pro flags issues in the argument, which would need rewriting by a human expert or a Lean formalization. The introduction of Tao's later manuscript (claim page) cites the write-up as claiming a slightly weaker statement than its Theorem 1.1 and records that it fell short of a complete proof. No revision was posted. The problem's standing takes nothing from this page.

Depends on. Nothing in this wiki.