What I thought about during today's midterm:
For any \( k \) there is a configuration of points and planes of type \( (3k_5) \), generalizing the \( (9_5) \). Simply place the points at the \( 2k \) vertices of a \( k \)-gonal prism, and at the \( k \) points where the diagonals cross in each of the prism's quadrilateral faces. There are \( k \) planes containing the quadrilateral faces, and \( 2k \) planes through each prism vertex and its two incident diagonals. The only requirement is that the prism be embedded in such a way that its quadrilateral faces are planar, but that still leaves a lot of degrees of freedom; for instance it can be knotted.