An Inequality for Finite Planar Spaces With No Disjoint Planes

We prove that for a finite planar space \( S = (\mathcal{P}, \mathcal{L}, \mathcal{H}) \) with no disjoint planes and with a constant number of planes on a line, the number \( \ell \) of lines is greater than or equal to the number \( c \) of planes, and the equality holds true if and only if \( S \) is either the finite Desarguesian 4-dimensional projective space \( PG(4,q) \), or the complete graph \( K_5 \).