Points with each pair having distance in range What is the maximum number of points can be placed on a plane such that the distance between any two is in some range? Specifically I'm interested in the range $[4,5]$, although I'm interested also in the general question. In addition, how does the question change if the setting is space ($R^3$) and not the plane?
I know that a maximum of three mutually equidistant points can be placed in the plane (four in space), which is a specific case where the range is just one number. For the case of between 4 and 5 I tried drawing rings around points to try to get some insight but I wasn't able to.
 A: For the case of the plane and the range $[4,5]$ I have tried to formalize the inelegant method of "drawing rings and seeing it's impossible to have 4 points" (I hope I wasn't wrong with the calculations of the inequalities): 
Without loss of generality one of the points is $(0,0)$ and another is $(a,0)$ for some $4\le a \le 5$. Any other point $(x,y)$ must satisfy that $4 \le x^2+y^2 \le 5$ and $4 \le (x-a)^2+y^2 \le 5$, therefore we have that $-1 \le a^2-2ax \le 1$ and so $\frac{a^2-1}{2a} \le  x \le \frac{a^2+1}{2a}$, so $\frac{15}{8}\le x \le \frac{13}{5}$. Then $x^2 \ge 3.5$ and so $ y^2 \le 1.5$. So we have $\frac{15}{8}\le x \le \frac{13}{5}$ and $y^2 \le 1.5$. Clearly the distance of any two such points is at most $4$, meaning it's impossible to have $4$ points with our restrictions, leaving us a maximum of $3$ points. 
I have a suspicion that the general answer is something like: the maximum number of points that can be placed in $R^n$ such that the distance between any two is in the range $(a,b)$, is equal to the the maximum number of points that can be placed in the line $R^1$ such that the distance between any two is in the range $(a,b)$ (trivially calculated) plus $n-1$. If correct the proof could be by induction, but I'm not sure how to exactly go about it.
