VC-dimension of spheres in 3 dimension I am searching for the VC-dimension of the following set system.

Universe $U=\{p_1,p_2,\ldots,p_m\}$ such that $U\subseteq \mathbb{R}^3$. In the set system $\mathcal{R}$ each set $S\in \mathcal{R}$ corresponds to a sphere in $\mathbb{R}^3$ such that the set $S$ contains an element in $U$ if and only if the corresponding sphere contains it in $\mathbb{R}^3$.

Details which I already know.


The VC-dimension is atleast 4. This is because if $p_1,p_2,p_3,p_4$ are 4 corners of a tetrahedron then it can be shattered by $\mathcal{R}$
The VC-dimension is atmost 5. This is because the set system can be embedded in $\mathcal{R}^4$ with spheres in $\mathcal{R}^3$ corresponding to hyperplanes in $\mathcal{R}^4$. It is known that hyperplanes in $\mathcal{R}^d$ have VC-dimension $d+1$.
My solution is incorrect. See other answer...