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Claim. For the $g_k(N)$ of Problem 866, with the $b_i$ distinct integers of which one may be non-positive (the paper's reading of the site's definition, Section 3, p. 2), W. van Doorn, The cardinality of a set containing the pairwise sums of a fixed number of integers, arXiv:2605.00040v1 (28 April 2026), proves Theorem 1, $g_3(N)=1$ for all $N\ge3$ (with $g_3(1)=g_3(2)=2$, and Theorem 2: integers $0\le b_1<b_2<b_3$ suffice); Theorem 3, $g_4(N)=3$ for all $N\ge2$, the lower bound from the odd integers together with $2N-2$ and $2N$; Theorem 5, $g_5(N)\ge4$ for $N\ge3$, from the odd integers together with $2N-4$, $2N-2$ and $2N$; and Theorem 8,

$$g_5(N)<1.2\cdot10^8\quad\text{for all }N$$

(the constant $113{,}591{,}719$, from the 1975 argument with explicit constants and a Sidon-set bound of O'Bryant), so that $g_5$ is bounded where the 1975 paper's $\log N$ lower bound holds only for the positive-integer variant $h_5$ (Theorem 4). The paper also states, with a proof sketch, Theorem 9, $g_k(N)\le h_k(N)<4N^{1-2^{2-k}}$ for $k\ge3$ and large $N$, a slight sharpening of the 1975 exponent, and $h_4(N)\le2270$ for the positive variant. Its Section 2 declares that ChatGPT (model GPT-5.3 Instant) was used for brainstorming and produced the proof of Theorem 3 on its own, and that the automated theorem prover Aristotle, from Harmonic, produced Lean formalizations of every statement marked with a checkmark, improving the $h_4$ bound on the way. The paper is compiled at its source card; nothing is independently reviewed.

Covers. The exact values $g_3(N)=1$ ($N\ge3$) and $g_4(N)=3$ ($N\ge2$), and the order of $g_5(N)$: bounded, with $4\le g_5(N)<1.2\cdot10^8$ for $N\ge3$; a later release claims the smaller constant $3{,}519{,}219$ on its own page. Not covered: the value of $g_5$ (the paper's Section 7 reports no counterexample to $g_5(N)\le5$ up to $N=15$ and does not exclude $g_5(N)\le4$ for large $N$); the constants for $k=6$, whose order is Choi, Erdős and Szemerédi's; the order of $g_k$ for $k\ge7$; and the exponent for large $k$, which Theorem 9 sharpens without closing.

Depends on. Nothing in this wiki. Theorem 8 follows the 1975 argument with explicit constants, but rests on the paper's own proof, not on the 1975 results recorded on their claim page.

Standing. Claimed. The paper is an arXiv preprint, on 2026-09-18 the only version, with no journal reference and no Crossref record, no citing paper and no independent review found. The site's curator thanks Wouter van Doorn on the problem page and credits van Doorn, in commentary last edited 1 December 2025 on a problem labeled OPEN, with $g_4(N)\le2032$, an earlier bound this paper supersedes; the commentary predates the paper and is not acceptance. Earlier postings of the same work by the author (the account Woett), both disclosed here and subsumed by the preprint: a note on the author's GitHub page, first posted on 30 August 2025 with the constant $2338$ and revised to $g_4(N)\le2032$ on 3 September 2025 (last revised 9 September 2025, the version linked above), linked from the site's thread, whose figure $2032$ the site's commentary adopted; and a thread comment of 26 February 2026 that made the two conventions explicit, showed that the 1975 example for $k=5$ fails for the site's $g_5$ with $b=(-1,2,3,5,6)$ when $N\ge6$, and proved $g_3(N)=1$ for $N\ge3$ in the comment itself. The thread comment of 4 May 2026 announced the preprint and linked the Lean file below. The page is named by the arXiv posting, the dated manuscript.

Formalization. Not counted as evidence. The paper's reference [5] is the author's own Lean file, ErdosProblem866.lean in the repository Woett/Lean-files, pinned above at its last change of 30 April 2026; its header names ChatGPT and Aristotle from Harmonic, and it states and proves $g_3(N)=1$ for $N\ge3$ (with $g_3(1)=g_3(2)=2$), $g_4(N)=3$ for $N\ge2$, $g_5(N)\ge4$ for $N\ge3$ and $g_5(N)<1.2\cdot10^8$ for all $N$, $h_3(N)=2$ for $N\ge4$, $h_4(N)\le2270$, $h_5(N)>\lfloor\log_2N\rfloor$ and the general bound of Theorem 9, together with explicit Sidon-set and weak Sidon-set bounds, with no sorry. The file carries six #print axioms commands but records no output. The corpus holds no build of it, so it gives no formalized evidence; no formal-conjectures statement file for the problem existed.