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on Erdős #662 · geometry / distances

The 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 $1$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.