Erdős Problem #657

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Distances are Euclidean. Distinct distances mean distinct numerical values; when a distance is said to occur a certain number of times, unordered pairs of distinct points are counted. Congruence allows any Euclidean isometry, including reflection. [S1] [S2]

Statement recorded in the supplied source.

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 $f(n)\to \infty$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove a stronger universal lower bound or settle the 1D equivalent sharply. [S1]

Short English statement

Must a planar set containing no isosceles triple determine more than every fixed linear multiple of its number of points in distinct distances?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2236 (EP-657), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 657, https://www.erdosproblems.com/657. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023). Bibliographic information from the supplied record.

• S4 Dumitrescu, Adrian, On distinct distances and {$\lambda$}-free point sets. Discrete Math. (2008), 6533--6538. Bibliographic information from the supplied record.

• S5 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. Bibliographic information from the supplied record.

• S6 Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. Bibliographic information from the supplied record.