On regular planar spaces of type (\(k,n\)) (Q2569928)

A planar space is a linear space given with a set of subspaces called planes such that through any 3 points not on a line there is a unique plane and it is the smallest subspace containing them; and every plane contains at least 3 points not on a line. A planar space is regular of type \((k,n)\) if every line has size \(k+1\) and every pencil of lines in a plane has size \(n+1\). The authors are interested in regular planar spaces of type \((k,n)\) with the additional property that any two planes intersect non-trivially. The projective spaces \(\text{PG}(3,n)\) and \(\text{PG}(4,n)\) are examples of such objects. They first prove that in such a space the number of lines is greater or equal to the number of planes. Then they show that if these two numbers are equal, the planar space is \(\text{PG}(4,n)\). To prove this, they use elaborate counting methods (using well-known designs equalities). Moreover they manage to reduce the problem to finding integral solutions to a degree two polynomial equation. They then check by computer that this equation has integral solutions only if \(k=n\) for \(n\leq 30000\). Whenever \(k=n\), the planar space must be \(\text{PG}(3,n)\) or \(\text{PG}(4,n)\). The authors conjecture this result true for any \(n\).