Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start:
- For $i\in\{1,\dots,n\}$:
- Sample a random point $X$ in the unit square.
- If $X$ is a distance at least $r$ from all points in $S$, add it to $S$.
My question is, what is the distribution of the size of $S$ after the process completes? I am interested in limiting behavior, e.g. setting $r=c/\sqrt{n}$ for fixed $c$ and letting $n\to\infty$.
