Erdős Problem #668

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 the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\to\infty$? Is it always $>1$ for $n>3$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine multiplicity for general n, with congruence rather than graph-isomorphism control. [S1]

Short English statement

Do increasingly large planar configurations maximizing unit-distance pairs admit unboundedly many incongruent optimal realizations?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2243 (EP-668), 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 668, https://www.erdosproblems.com/668. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 B. Alexeev, D. Mixon, and H. Parshall, The Erdős unit distance problem for small point sets. arXiv:2412.11914 (2025). Bibliographic information from the supplied record.

• S4 P. Engel, O. Hammond-Lee, Y. Su, D. Varga, and P. Zsámboki, Diverse beam search to find densest-known planar unit distance graphs. arXiv:2406.15317 (2025). Bibliographic information from the supplied record.