Claims

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2008_12_01_dumitrescu: Dumitrescu's 2008 note proves the problem's assertion for collinear sets: n points on a line with no isosceles triple determine at least (log n)^c n distinct distances, and some determine at most n 2^{O(sqrt(log n))}; refereed.

2025_08_19_hunter: An observation credited to Zach Hunter in the site's commentary, with details by Alfaiz and Tang in the thread, that collinear sets with no isosceles triple determine at least 2^{c (log n)^{1/9}} n distances; pending.

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