If I have a disk $d$ where each point of the disk is contained in at least $k$ other disks, then at least how many other disks does $d$ intersect?
Given, that all the disks (including $d$) have the same radius, and no two disks have the same coordiante.
A simple upper limit is $3k$. Each disk passes through the center of $d$ and is clocked from the first one by $\frac{2\pi}{3k}$. If one of the covering disks is allowed to match $d$ we get $3k-2$. But one can probably do better.