An Inequality for Finite Planar Spaces With No Disjoint Planes

Abstract

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 $K_5$.