Given a discrete point set $S$ in ${\bf R}^2$ with a specified base-point $p_0 \in S$, label the remaining points as $p_1, p_2, \dots$ in order of increasing distance from $p_0$ (with ties resolved indifferently), and let $d_n(S,p_0)$ be the distance between $p_0$ and $p_n$.

What is known about the infimum of $d_n(S,p_0)$ as $S$ and $p_0$ vary, if $S$ is required to have the property that no two of its points are less than 1 apart? An equivalent statement of the problem replaces each point-set with a packing of the plane by disjoint disks of radius 1/2. It would be very nice if the infimum of $d_n$ was achieved by the hexagonal packing of the plane, but my intuition says that for some $n$ this is not the case; I wonder if this is known.