# Placing $n$ points so that their distances lie in $[1,a]$

What is the maximum number of points we can place in the plane so that the distance between any two such points is in the interval $[1,a]$?

I had initially conjectured that the maximum could be found by trying with regular $n$-gons of side $1$ until the largest diagonal exceeded $a$.

However I have recently found this to be false. Placing $7$ points with distances in $[1,2]$ is possible by taking a hexagon of side length $1$ and placing one point on each vertex and one in the center of the hexagon. On the other hand a heptagon of side length $1$ has its largest diagonals of length approximately $2.246$.

What is the maximum number of points we can place given $a$? Is there a neat construction which achieves this all the time?

Also note that if we can solve this problem then we can solve the problem of placing $n$ Points on the plane so as to maximize the number of pairs of points $(a,b)$ at distance between $1$ and $a$. This is because if we know the maximum possible clique we can build the Turan Graph by placing a lot of points in the same place.

Anyway, some heuristics are quite easy. If we take $n$ points inside a unit circle, with respect to a uniform probability distribution, we expect that the most distant points are two units apart and the infimum of the distances is around $\frac{2}{\sqrt{n}}$.
So, assuming that $a$ is big, we expect to be able place around $a^2$ points in such a way that their distances lie in $[1,a]$, but not much more.