Two circles in 3-D are linked iff each one passes through the interior of the other.

There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique circle passing through the points.

Does some $N$ guarantee that there must be a linked pair among the $N \choose 3$ circles? If so, what is the smallest such $N$?

(Circles sharing one of the $N$ points do not count. There are ${N \choose 3}{N-3 \choose 3}/2 = {N \choose 3,3,n-6}/2$ eligible pairs of circles.)

I have no idea whatever how to go about finding $N$ (if there is one). If I had time I would write a Mathematica program to look into this.