$S$ is the $3$-dimensional sphere (in $\mathbb R^3$) and $T$ is a triangulation of $S$. It is known that $T$ can be extended to a triangulation of the ball by adding points inside the ball (one point at each step), and connecting each to at most $6$ previous points.
I need to give an example of a tringulation of the sphere for which adding points connected to $5$ previous points does not yield an extension to a triangulation of the ball.
I can not think of such a triangulation, so any help would be appreciated.
An answer is a triangluation $T$ formed by the vertices of a regular icosahedron inscribed in the sphere.
Since every vertex in $T$ has five emerging faces, if you add an interior point connected to fewer than six vertices, you make no progress in reducing the minimum number of additional points needed to complete the triangularization.