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Claim. Theorem 2 of Lee, Pohoata and Zhu, The Minkowski grid has robustly many repeated distances, arXiv:2607.05374v1 (6 July 2026), gives an absolute constant $\delta>0$ and, for every positive integer $n$, a set $P$ of $n$ points in $\mathbb{R}^2$ in which every subset $A\subseteq P$ with $|A|\ge2$ has some distance occurring $\gtrsim|A|^2/n^{1-\delta}$ times among its ordered pairs. A subset with all distances distinct has each distance only twice, so its Corollary 3(3) deduces that, for all large $n$, every subset of $P$ with at least $n^{1/2-\delta}$ points determines a repeated distance. Hence $F_2(n)\lesssim n^{1/2-\delta}$, where $F_2(n)$, the paper's $f_{\mathrm{dist}}(n)$, is the function of Problem 1208 for $d=2$. The paper says that this answers negatively the question of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky whether $F_2(n)=n^{1/2-o(1)}$, and that the previous best bound was $F_2(n)\lesssim n^{1/2}\exp(-c\log n/\log\log n)$, from the grid estimate of Croot, Mao, Pohoata, Sheffer and Yip (their Theorem 1.3). The set is a Minkowski grid built from a totally real number field of high degree, the construction that also answers the question of Problem 1207. The paper's acknowledgments say that the collaboration started from the first author's manuscript, in which he used ChatGPT to find a set with this property, recorded on its claim page, and that the new point of the joint note is to obtain both bounds from one construction with a simpler analysis. The statements are read from the PDF of version 1 (pp. 2--3); the proof was not checked here.

Covers. An upper bound in the case $d=2$: $F_2(n)\lesssim n^{1/2-\delta}$. It does not determine the order of $F_2(n)$, and nothing is claimed for $d\ge3$.

Depends on. No page of this wiki. The paper uses as a black box its Proposition 4, a mild strengthening of Proposition 2.3 of Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood (arXiv:2605.20695, 2026), on towers of totally real fields.

Standing. Claimed. The result is an arXiv preprint (version 1 of 6 July 2026) with no journal publication and no Lean proof. The erdosproblems.com remarks (page last edited 7 September 2026) credit the bound $F_2(n)\ll n^{1/2-c}$ to this paper, but Bloom's catalog labeled the problem OPEN, and commentary on an open problem is not acceptance.