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Claim. The answer to the limit question of Problem 1089 is yes, and the estimate is exact to leading order. Theorem 5 of Feng et al. states that $g_d(1)=2$ and that, for every $n\ge2$, the limit exists and equals $1/(n-1)!$; its proof shows that

$$\binom{d+1}{n-1}+1\le g_d(n)\le\binom{d+n-1}{n-1}+1 \qquad\text{so}\qquad \lim_{d\to\infty}\frac{g_d(n)}{d^{n-1}}=\frac{1}{(n-1)!}.$$

The proof first notes that $g_d(n)-1$ is the largest size of a set in $\mathbb{R}^d$ determining at most $n-1$ distinct nonzero distances, an $(n-1)$-distance set. The upper bound is Theorem 1 of Bannai, Bannai and Stanton (Combinatorica 3 (1983), 147–152), which bounds an $s$-distance set in $\mathbb{R}^d$ by $\binom{d+s}{s}$ points, applied with $s=n-1$. The lower bound is a construction: the $0/1$ vectors of length $d+1$ with exactly $n-1$ ones lie in a hyperplane isometric to $\mathbb{R}^d$, number $\binom{d+1}{n-1}$, and two of them sharing $k$ ones are at squared distance $2(n-1-k)$, so at most $n-1$ distances occur. Both binomial coefficients are $d^{n-1}/(n-1)!+O(d^{n-2})$, which gives the limit. The construction generalizes the one for two distances recorded on Problem 502, whose question, the exact size of the largest two-distance set, is the case $n=3$ of the exact estimate and remains open.

Depends on. [[problems/distance_problems/E0502/claims/1983_06_01_bannai_bannai_stanton|The Bannai–Bannai–Stanton bound]], an accepted claim, for the upper bound; the lower construction and the limit rest on no other page.

Claimant. The result is Section 4.4 of the preprint by Tony Feng and twenty-three coauthors, arXiv:2601.22401, first posted 2026-01-29 and carded as [[../library/distance_problems/feng_2026_semi_autonomous_mathematics_discovery_gemini_case/_index|Feng et al. 2026]] (version 3 of 2026-02-05, linked above). The paper reports that the solution was produced by Aletheia, a research agent built on Gemini Deep Think, and that the authors edited only the upper-bound part, replacing the agent's sketch of the Bannai–Bannai–Stanton argument by a citation of their theorem. The paper classifies the result as an independent rediscovery: its Remark 4.4 reports that the authors' human experts found the problem already answered in Remark 3(ii) of Bannai and Bannai, Combinatorica 1 (1981), 99–102, whose authors appear not to have connected their remark to Erdős's question, and notes that the agent's trace cites the 1983 sequel many times without accessing the 1981 paper. The remark adds that, although this meets the paper's definition of an independent rediscovery, the authors judge it a very likely case of what they call subconscious AI plagiarism. The attribution to the 1981 remark is cited from Remark 4.4 of the paper.

Acceptance. Thomas Bloom, the site's curator, marks the problem solved and credits, on the problem page last edited 2026-02-01, the lower bound to Aletheia as a generalization of the Problem 502 construction and the upper bound to Bannai, Bannai and Stanton, stating the resulting limit. The preprint has no journal publication, so the claimant's own result is not refereed; the upper bound it cites is a refereed theorem of 1983. The claim is accepted on the curator's documented acceptance.

Formalization. Boris Alexeev's repository of formalized Erdős problems holds a Lean 4 file, linked above at the commit that the formal-conjectures statement file pins and first added on 2026-08-17, that declares itself a formalization of this solution with Bannai, Bannai, Stanton and Aletheia as its informal authors and the AI systems Codex and GPT-5.6 Sol as its formal authors. Its theorem Erdos1089.erdos_1089 states both bounds and the limit for $n\ge2$. The development is not built or audited in this repository, so it gives no formalized evidence.

What remains. The exact value of $g_d(n)$, equivalently the largest size of an $(n-1)$-distance set in $\mathbb{R}^d$, is open for every $n\ge3$ beyond small cases; the two bounds differ in their lower-order terms. The function is the inverse of the $f_d$ of Problem 1083, which asks about the other direction of the same relation.