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Claim. Adam Sheffer's survey Distinct Distances: Open Problems and Current Bounds, arXiv:1406.1949, first posted 8 June 2014 (v3 of 2 July 2018), writes $\phi(n,k,\ell)$ for the least number of distinct distances among $n$ planar points of which every $k$ determine at least $\ell$ distances. Its table of bounds lists $\phi(n,4,3)=O(n/\sqrt{\log n})$, the answer yes to Problem 659. Its section on local properties reads the condition as excluding squares only and takes the $\sqrt n\times\sqrt n$ section of the triangular lattice, which determines $\Theta(n/\sqrt{\log n})$ distances and contains no square. The source card is sheffer_2014_distinct_distances_open_problems_current_bounds.

Why the claim is rejected. Four points determine only two distances exactly when they are similar to one of six configurations, and the square is only one of them. The triangular lattice contains others, such as the rhombus made of two equilateral triangles, so it violates the condition and the argument fails. Terence Tao pointed this out on the site's discussion thread on 13 January 2026. The bound itself is true: Grayzel's Theorem 1 proves it, as Grayzel's claim page records. The rejection concerns the survey's proof, not its statement.

Depends on. No page of this wiki.