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Claim. Three collinear points determine two equal distances exactly when the middle one is the midpoint of the other two, so a set on a line has no isosceles triple exactly when it contains no nontrivial three-term arithmetic progression, and such a set is an admissible set for Problem 657. Its distinct distances are the positive differences of its elements, $(|A-A|-1)/2$ in number. Dumitrescu proves that every progression-free set of $n$ reals has at least $(\log n)^c\,n$ distinct differences, for an absolute constant $c>0$, and that some such sets have at most $n\,2^{O(\sqrt{\log n})}$. Writing $f(n)$ for the least number of distinct distances of an admissible collinear $n$-set divided by $n$, the two bounds read $(\log n)^c\le f(n)\le2^{O(\sqrt{\log n})}$, as the site's remark states them. The lower bound answers the problem's question yes for collinear sets, a case Erdős called open even on the line in [Er73]; the upper bound shows that for collinear sets $f(n)$ grows no faster than $2^{O(\sqrt{\log n})}$, and, since collinear sets are planar sets, that the planar quantity grows no faster either. The paper is A. Dumitrescu, On distinct distances and $\lambda$-free point sets, Discrete Math. 308 (2008), no. 24, 6533–6538, DOI 10.1016/j.disc.2007.11.046; the publisher's record dates the issue to December 2008 and the page name carries the first day of that month, since the record gives no day. The paper is not held in the library, and the bounds are stated here as the site's remark credits them.

Covers. The problem's assertion for sets on a line: an $n$-point collinear set with no isosceles triple determines at least $(\log n)^c\,n$ distinct distances, so $f(n)\to\infty$ on that class, and the class contains sets with at most $n\,2^{O(\sqrt{\log n})}$ distances. Not covered: planar sets not on a line, for which no superlinear lower bound is known; the problem's question remains open.

Depends on. No page of this wiki.

Acceptance. The paper is refereed: it appeared in the journal Discrete Mathematics, volume 308 (2008). The site labels the problem OPEN, and its remark (page last edited 15 October 2025) credits Dumitrescu [Du08] with the two bounds; that remark on an open problem is not an acceptance, so no reviewed evidence is listed. The lower bound was later raised to $2^{c(\log n)^{1/9}}$ on Hunter's claim page.