Given a positive real number $R$ and $n$ fixed points in a plane, find the largest possible value $M$ such that if the pairwise distance between the $n$ points is less or equal to $M$, there exists a circle of radius $R$ that contains all $n$ points.
I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $M$, then there is definitely a circle of radius $R$ that encloses all of them, where this M is as large as possible.
Obviously, for two points, the answer is $2R$. For three points, it gets more complicated, but using some geometry, I believe the answer is $\sqrt{2}R$ (as in $\sqrt{R^2 + R^2}$), but I haven't been able to prove it. I would like to generalize this to $n$ points, where I am fine with the answer being dependent on $n$.
My hypothesis is that $\sqrt{2}R \leq M \leq 2R$ for all $n$, but I would be mainly interested in how large $M$ can be made as the number of points increases.