Erdős Problem #662 — Consider the triangular lattice with minimal distance between two points $1$ .

About 29 years old · traced to 1997

Consider the triangular lattice with minimal distance between two points $1$ . Denote by $f(t)$ the number of distances from any points $\leq t$ . For example $f(1)=6$ , $f(\sqrt{3})=12$ , and $f(3)=18$ . Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that $d(x_i,x_j)\geq 1$ for all $i\neq j$ . Is it true that, provided $n$ is sufficiently large depending on $t$ , the number of distances $d(x_i,x_j)\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice? In particular, is it true that the number of distances $\leq \sqrt{3}-\epsilon$ is less than $1$ ?

References

Primary source

ULAM AI, UnsolvedMath dataset, Erdős Problems set, revision b9437975f3c873f635a13c48f8b022f5ba80898a; CC BY 4.0.

Additional references

UnsolvedMath, Erdős Problems set, ULAM AI, licensed CC BY 4.0.

distances geometry

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Open

No proof or counterexample has been found, and the problem’s published wording appears defective.

Erdős, Lovász, and Vesztergombi ask whether sufficiently large one-separated point sets in the plane can have more pairs at distance at most $t$ than the corresponding triangular-lattice configuration, for every threshold $t$ . The original formulation appears to contain numerical errors, and its stronger literal reading is described as nonsensical.

Known results

Fejes Tóth, 1942: hexagonal packing is optimal for the related densest-packing problem; this does not establish the full threshold-by-threshold extremality claim.
The discussion proposes corrected triangular-lattice counts such as $f(2)=18$ , $f(\sqrt{7})=30$ , and $f(3)=36$ , but supplies no proof of the conjectured extremality.

Current status (as of March 2026): The exact short-distance extremality problem remains open; only related packing results and clarification of defects in the formulation are recorded.

Sources

No solutions have been posted yet.