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OP-9120OpenGeometry and topology

Erdős #662 · geometry / distances

Consider the triangular lattice with minimal distance between two points 11. Denote by f(t)f(t) the number of distances from any points ≤t\leq t. For example f(1)=6f(1)=6, f(3)=12f(\sqrt{3})=12, and f(3)=18f(3)=18.

Let $x_1,\ldots,x_n…

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Consider the triangular lattice with minimal distance between two points 11. Denote by f(t)f(t) the number of distances from any points ≤t\leq t. For example f(1)=6f(1)=6, f(3)=12f(\sqrt{3})=12, and f(3)=18f(3)=18.

Let x1,…,xn∈R2x_1,\ldots,x_n\in \mathbb{R}^2 be such that d(xi,xj)≥1d(x_i,x_j)\geq 1 for all i≠ji\neq j. Is it true that, provided nn is sufficiently large depending on tt, the number of distances d(xi,xj)≤td(x_i,x_j)\leq t is less than or equal to f(t)f(t) with equality perhaps only for the triangular lattice?

In particular, is it true that the number of distances ≤3−ϵ\leq \sqrt{3}-\epsilon is less than 11?

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Background

A problem of Erdős, Lovász, and Vesztergombi.

This is essentially verbatim the problem description in [Er97e], but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.

Erdős also goes on to write 'Perhaps the following stronger conjecture holds: Let t1<t2<⋯t_1<t_2<\cdots be the set of distances occurring in the triangular lattice. t1=1t_1=1 t2=3t_2=\sqrt{3} t3=3t_3=3 t4=5t_4=5 etc. Is it true that there is an ϵn\epsilon_n so that for every set y1,…,y_1,\ldots, with d(yi,yj)≥1d(y_i,y_j)\geq 1 the number of distances d(yi,yj)<tnd(y_i,y_j)<t_n is less than f(tn)f(t_n)?'

Again, this is nonsense interpreted literally; I am not sure what Erdős intended.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #662 link

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