I have encountered this question as a part of some other research. The problem appears to be a combinatorial problem. But my knowledge of results in combinatorics and discrete math is quite thin. I was hoping someone here may know the answer.

Let $X$ be a finite set $(X,d)$ be a discrete metric space. Form subsets $A$ of a given size $k$ by drawing elements from $X$ with the restriction that for any $x,y \in A$, the distance $d(x,y) \leq \alpha$, where $\alpha$ is a given. Assume that the values of $\alpha$ and $k$ are not degenerate: assume there do exist sets $A$ with diameter $\alpha$ and size $k$ each so that their union is $X$. Also assume $\alpha$ is smaller than the diameter of $X$. What then is the maximum number of disjoint sets $A$ with this diameter and size? Or are there bounds in terms of $|X|,k,d,\alpha$?

Clearly, if I remove the restriction of the distance and allow *arbitrary* sets $A$, the answer is trivial $=\lfloor|X|/k\rfloor$. The distance restriction makes it harder.

I would be happy if someone can at least point me to an suitable body results which could be useful in this problem.