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Claim. Let $p$ be prime and let $A\subseteq\mathbb{F}_p$ be a uniformly random subset of size $k=\log_2p+o(\log\log p)$. Then the probability that every element of $\mathbb{F}_p$ is a subset sum of $A$ tends to $0$ as $p\to\infty$ (Theorem 2.1 of the note). Hence $f(p)>\log_2p+o(\log\log p)$ along the primes, and the bound $f(N)\le\log_2N+o(\log\log N)$ asked for in Problem 543 fails, as Erdős expected. The claim is the note's qualitative statement; the quantitative lower bound $f(p)\ge\log_2p+(\tfrac1{2\log2}+o(1))\log\log p$ of Ma and Tang's later paper is a separate, pending claim, Ma and Tang 2026.

Source. Q. Tang, A note on Problem #543, posted to the site's forum on 2026-01-21 and revised on 2026-01-23 (the two GitHub links, each pinned to its commit); the note's byline names Tang and ChatGPT-5.2 Pro, and Tang wrote in the thread that they thought they might have disproved the problem using ChatGPT and, with the revision, that they had gone through the AI-generated draft step by step and corrected the places needing more justification. The repository's README says that Ma and Tang's paper supersedes the note; that paper has its own page. The claimant is the human submitter; the AI system is named as the note's byline gives it.

The argument. The note compares the uniform $k$-subset with $k$ independent uniform elements, which differ by $o(1)$ in probability since $k=O(\log p)$, and shows that with $2^k=p\,(\log p)^{o(1)}$ some element of $\mathbb{F}_p$ is missed by every subset sum with probability tending to one; the unrepresented element is found through the distribution of the number of representations, in the spirit of Erdős and Hall. The context is the upper bound $f(N)\le\log_2N+O(\log\log N)$ of Erdős and Rényi 1965 (Theorem 2) and the weaker obstruction of Erdős and Hall 1978 (Theorem 2), which rules out $o(\log\log\log N)$ for cyclic groups.

Acceptance. Reviewed: the site's curator, Thomas Bloom, labels the problem disproved on the problem page and credits ChatGPT and Tang (page last edited 2026-01-27; the community database records the disproved status from 2026-01-23); Bloom's own comment in the thread on 2026-01-21 says only that the note sounds believable at a quick skim and that they would search the literature. In the same thread on 2026-01-21, Terence Tao tentatively assessed the argument as correct while noting that a human-written proof or a Lean verification would still be desirable; that documented assessment by a named expert supports the curator's acceptance. Not refereed: the note has no journal publication. No Lean formalization is recorded, and this corpus has not independently verified the proof.

Depends on. No page of this wiki; the result rests on the cited note.