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Claim. Theorem 2 of Lee, Pohoata and Zhu 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 $|A|\ge 2$ has some distance repeated $\gtrsim |A|^2/n^{1-\delta}$ times. The set is a Minkowski grid built from a totally real number field of high degree. Their Corollary 3(2) deduces that, for all large $n$, every subset of $P$ with at least $n^{1-\delta}$ points contains an isosceles triangle. The paper counts three equally spaced collinear points as a degenerate isosceles triangle, which matches the site's reading of the case $d=1$ as the three-term progression problem. Hence $P_2(n)\lesssim n^{1-\delta}$, so $P_2(n)<n^{1-c}$ for every $0<c<\delta$ and all large $n$. The paper says that this application confirms the conjecture of Erdős in [Er80, p. 110].

Covers. The particular question of Problem 1207, whether $P_2(n)<n^{1-c}$ for some constant $c>0$: yes. The estimate of $P_d(n)$ in general, including the right exponent for $d=2$, is not settled.

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, Remarks on the disproof of the unit distance conjecture (arXiv:2605.20695, 2026), on towers of totally real fields; that paper bears on Problem 90.

Standing. Claimed. The result is an arXiv preprint (v1 of 6 July 2026) with no journal publication and no Lean proof. The site's remarks, edited 7 September 2026, credit the construction to Lee, Pohoata and Zhu, assisted by ChatGPT, and say that it answers the main question of Erdős; the site still labels the problem OPEN, and commentary on an open problem is not acceptance. The paper's acknowledgment records that ChatGPT helped the first author find the related construction of a set in which every subset of at least $n^{1/2-\delta}$ points determines a repeated distance.