{"schema":"vela.problem-packet.v0.1","problem":662,"statement":"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 $1$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}