Classes of circle rims
In fact, the VC dimension is 3.

To show this, we need to show that there is a set of 3 points which is shattered, and that no set of 4 points is shattered.

All sets of 3 points behave the same, but for definiteness let us choose three points $x,y,z$ which form the corners of an equilateral triangle. By symmetry, there are only 4 cases to consider:


$\emptyset$: choose some arc which avoids all points.
$\{x,y,z\}$: choose the complement arc.
$\{x\}$: choose a small arc around $x$.
$\{y,z\}$: choose the complement arc.


This shows that the VC dimension is at least 3.

To show that the VC dimension is at most 3, let us consider some arbitrary set of points $x,y,z,w$, presented in clockwise order starting at an arbitrary point. An arc that contains both $x$ and $z$ must contain either $y$ or $w$, and in particular there is no arc that contains exactly $\{x,z\}$.