Problem 657

Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\to \infty$ ?

Source: erdosproblems.com/657

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 15 October 2025). The assertion is proved for collinear sets, which are the progression-free sets of reals: Dumitrescu's accepted partial claim gives $(\log n)^c\le f(n)\le2^{O(\sqrt{\log n})}$ on that class, and Hunter's pending partial claim raises the lower bound to $2^{c(\log n)^{1/9}}$ . No superlinear lower bound is known for planar sets in general, so the standing derived from the claim pages is open.

BlSi23: T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).
Du08: Dumitrescu, Adrian, On distinct distances and $\lambda$ -free point sets. Discrete Math. (2008), 6533-6538.
Er73: Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
Er97e: Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
KeMe23: Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).

Problem 657

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Proof claims

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