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.