---

Claim. Let $f(n)$ be, as on [[problems/distance_problems/E0657/claims/2008_12_01_dumitrescu|Dumitrescu's claim page]], the least number of distinct distances of an $n$-point collinear set with no isosceles triple, divided by $n$; such sets are exactly the sets of $n$ reals without a nontrivial three-term arithmetic progression, and they are admissible sets for Problem 657. The site's commentary credits Zach Hunter with the observation, made in the problem's discussion thread on 19 August 2025, that a result of Ruzsa combined with standard tools of additive combinatorics turns the recent bounds on the size of progression-free subsets of $\{1,\ldots,N\}$ into the lower bound

$$f(n)\ge2^{c(\log n)^{1/9}}$$

for an absolute constant $c>0$, a quasipolynomial improvement of Dumitrescu's $(\log n)^c$. The details were written out in the thread by Alfaiz on 1 October 2025, using the Kelley–Meka bound (card) as sharpened by Bloom and Sisask (card): a progression-free set with few distinct differences would, by Ruzsa's argument and the Plünnecke–Ruzsa inequalities, embed densely into a short interval, where those bounds forbid it. Quanyu Tang's reply of the same day noted, and Alfaiz agreed on 2 October 2025, that the deduction holds for subsets of the line or of a torsion-free abelian group and not for planar sets, since a projection does not preserve the absence of isosceles triangles. A later thread comment by Alfaiz, of 6 August 2026, states that Raghavan's 2026 progression-free bound raises the exponent $1/9$ to $1/6$; that comment is disclosed here and gets no page of its own.

Covers. The problem's assertion for sets on a line, with the bound $f(n)\ge2^{c(\log n)^{1/9}}$. Not covered: planar sets not on a line; the problem's question remains open.

Depends on. No page of this wiki.

Standing. Claimed. The result exists as a thread observation and the site's commentary: no written source by its authors states it, and it has no arXiv version, no journal record, no formalization and no independent review. The commentary credits it to Hunter, with details by Alfaiz and Tang, but the site labels the problem OPEN (page last edited 15 October 2025), so the credit is not an acceptance and no reviewed evidence is listed. The claim is dated by Hunter's comment of 19 August 2025.