I hope that you can help me with this.
Let P be a set of points in the plane, such that $|P|=n$, what is the maximal number of open disks containing at least $k$ points for some $k$, two discs are equivalent if they contain the same points.
I have some intuition here, but I'm not sure if I should follow it.
the number of distinct open discs containing at least $k$ points, for $k>2$ is bounded by ${n\choose 3}$, since every disk is uniquely defined by the 3 points closed to its boundary. Every 3 points form a triangle bounded by some disk. Suppose two different disks have the same 3 points being closest to the edge, than at least one disk has a point contained in it, which is not contained inside the other disc, than we can "shrink" the first disk until the "spare" point is closest to its edge, then it is defined by a different triplet.
Is there any flaw in my thinking?