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Claim. Sungchul Lee, Planar Point Sets with Small Distinct-Distance Subsets, a manuscript of 14 June 2026 in the author's repository Erdos1208 (the preprint link, pinned to the repository's last revision of that day), states as Theorem 1.1 that there is an absolute constant $\varepsilon>0$ with

$$F_2(n)\ll n^{1/2-\varepsilon},$$

where $F_2(n)$ is the function of Problem 1208 for $d=2$: every set of $n$ points in the plane contains $F_2(n)$ points with all pairwise distances distinct. This is a fixed power below the square-grid bound $F_2(n)\ll n^{1/2}/(\log n)^{1/4}$. The manuscript builds its point sets from boxes in the rings of integers of a tower of totally real number fields of growing degree and bounded root discriminant, the arithmetic input of the OpenAI unit-distance construction, in which fixed primes $q\equiv1\pmod4$ split completely. It lifts to them a split-prime sieve of Pohoata and Sheffer that bounds the distinct-distance subsets of the integer grid by $N\exp(-c\log N/\log\log N)$; each split prime contributes one sign choice per prime ideal, and that saving, exponential in the degree, becomes a fixed power of $n$. Its section "Use of AI" says that the author used OpenAI's GPT-5.5 Pro to explore proof strategies and obtain ideas, and then verified the details himself. The statement is read from the TeX source at the pinned revision; the proof was not checked here.

Submission note. Posted to the erdosproblems.com forum by Sungchul Lee on 14 June 2026:

With the help of GPT-5.5 Pro, I have obtained a polynomial improvement to the upper bound for this problem:

$$> F_2(n) \ll n^{1/2-\varepsilon} >$$

for some absolute constant $\varepsilon>0$. This improves the previous square-grid upper bound $F_2(n)\ll n^{1/2-o(1)}$.

The manuscript is available here: https://github.com/lsngchl/Erdos1208

The main idea is to combine the number-field towers of bounded root discriminant used in the recent OpenAI unit-distance construction with a split-prime argument of Pohoata and Sheffer.

I have checked the mathematical details of the proof myself, but I am not an expert in this area, so I would be very grateful for any comments, corrections, or pointers to relevant prior work.

Covers. An upper bound in the case $d=2$: $F_2(n)\ll n^{1/2-\varepsilon}$. 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 manuscript takes its number-field tower from the OpenAI unit-distance construction and from Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood (arXiv:2605.20695, 2026), and its grid sieve from a blog post of Pohoata and Sheffer.

Standing. Claimed. The manuscript is self-published, with no journal publication and no Lean proof. In the erdosproblems.com thread, Nat Sothanaphan reported on 14 June 2026 a screening check that found no error, which is not a review of the proof. The erdosproblems.com remarks credit the bound to the later joint paper of Lee, Pohoata and Zhu, recorded on its claim page, whose acknowledgments say that this manuscript first established it.