Let N be a set of n arbitrary points in $\mathbb{R}^2$ and let $k \in \{1,...,n\}$. Is it true that there always exists a disk $D$ containing exactly k points of N?



Existence of a disk containing a given number of points of a discrete set
Yes. From a generic point (in this case, in the complement of the ${n \choose 2}$ perpendicular bisectors of each pair), the distances to the members of $N$ are all
distinct, so for each $k$, disks of suitable radius centred at such a point will contain exactly $k$ of them.


