Bounding the size of certain rank 3 geometries with designs as rank 2 residues

Abstract

We consider locally finite geometries of rank 3 belonging to certain diagrams where all strokes represent classes of non-trivial designs, at least one of them consisting of symmetric designs other than projective planes. The gonality diagram of such geometry is of spherical type, whereas the diameter diagram is of affine type. In cases like this, the criteria available from 24 cannot tell us anything about the finiteness or infiniteness of a geometry. In the main theorem of this paper we prove that, certain imply finiteness additional conditions on the parameters of the designs associated with the strokes of the diagram, a geometry is necessarily finite.