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Claim. The answer to Problem 135 is no. Theorem 1.2 of Tao gives, for every sufficiently large $L$, a subset $A_L$ of the grid $\{0,\ldots,L-1\}^2$ with $|A_L|\gg L$ avoiding the eight four-point patterns that Dumitrescu classified as the ways four points can fail to determine five distinct distances. Thinning $A_L$ to exactly $n$ points with $L\asymp n$ gives, for every large $n$, a set of $n$ plane points in which every four points determine at least five distinct distances while the whole set, lying in a grid with $O(L^2/\sqrt{\log L})$ distances, determines

$$\ll \frac{n^2}{\sqrt{\log n}}$$

distinct distances, so such a set need not determine $\gg n^2$ distances. The same sets refute the stronger conjecture of Erdős on the problem page, that such a set contains $\gg n$ points with all pairwise distances distinct. The construction randomizes the finite-field parabola construction of Erdős and Turán, which already avoids the parallelogram pattern, and combines it with Dumitrescu's probabilistic treatment of the other seven patterns. The paper is carded at tao_2024_planar_point_sets_forbidden_4_point, which records the theorem, its consequence for this problem and the general position of the sets (Remark 1.8).

Depends on. No page of this wiki.

Acceptance. The paper is refereed: Terence Tao, Planar point sets with forbidden 4-point patterns and few distinct distances, Discrete Comput. Geom. 76 (2026), no. 1, 643–651, published online 2025-10-28; the preprint arXiv:2409.01343 was first posted on 2024-09-02 and discussed by the author on Tao's blog the next day. The site's curator, Thomas Bloom, marks the problem disproved and credits Tao's construction on the problem page; the site's export of 2026-09-04 records the label "DISPROVED (LEAN)". No proof was checked here.

Formalization. Boris Alexeev's repository of formalized Erdős problems holds a Lean development, added on 2026-08-17 and linked above at a pinned commit, whose header names Tao as informal author and Codex and GPT-5.6 Sol as formal authors; it constructs, for every $n$, exactly $n$ points of the Euclidean plane in which every four determine at least five distances while the total number of distances is $O(n^2/\sqrt{\log n})$. No formal-conjectures statement of the problem is recorded. The development was not built or audited here, so no formalized evidence is listed.